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ON DEDEKIND’S “UBER DIE PERMUTATIONEN DES KORPERS

ALLER ALGEBRAISCHEN ZAHLEN”

By

Joseph J.P. Arsenault, Jr.

B.S. University of Maine, 1995

A THESIS

Submitted in Partial Fulfillment of the

Requirements for the Degree of

Master of Arts

(in Mathematics)

The Graduate School

The University of Maine

August 2015

Advisory Committee:

William M. Snyder, Professor Emeritus of Mathematics, Co-Advisor

Eisso J. Atzema, Lecturer in Mathematics, Co-Advisor

Andrew H. Knightly, Associate Professor in Mathematics

LIBRARY RIGHTS STATEMENT

In presenting this thesis in partial fulfillment of the requirements for an advanced

degree at The University of Maine, I agree that the Library shall make it freely available

for inspection. I further agree that permission for “fair use” copying of this thesis for

scholarly purposes may be granted by the Librarian. It is understood that any copying

or publication of this thesis for financial gain shall not be allowed without my written

permission.

Signature: Date:Joseph J.P. Arsenault, Jr.

ON DEDEKIND’S “UBER DIE PERMUTATIONEN DES KORPERS

ALLER ALGEBRAISCHEN ZAHLEN”

By Joseph J.P. Arsenault, Jr.

Thesis Co-Advisors: Dr. William M. SnyderDr. Eisso J. Atzema

An Abstract of the Thesis Presentedin Partial Fulfillment of the Requirements for the

Degree of Master of Arts(in Mathematics)

August 2015

We provide an analytic read-through of Richard Dedekind’s 1901 article “Uber

die Permutationen des Korpers Aller Algebraischen Zahlen,” describing the principal

results concerning infinite Galois theory from both Dedekind’s point of view and a

modern perspective, noting an apparently uncorrected error in the supplement to the

article in the Collected Works. As there is no published English-language translation of

the article, we provide an annotated original translation.

THESIS ACCEPTANCE STATEMENT

On behalf of the Graduate Committee for Joseph J.P. Arsenault, Jr., we affirm

that this manuscript is the final and accepted thesis. Signatures of all committee

members are on file with the Graduate School at the University of Maine, 42 Stodder

Hall, Orono, Maine.

Signatures: Date:Dr. William M. SnyderProfessor Emeritus of Mathematics

Date:Dr. Eisso J. AtzemaLecturer in Mathematics

ii

TABLE OF CONTENTS

LIST OF TABLES ....................................................................................... iv

LIST OF FIGURES....................................................................................... v

Chapter

1. INTRODUCTION ..................................................................................... 1

2. MODERN EXPOSITION OF RESULTS AND SYNOPSIS OF ARTICLE ........... 3

2.1. Introduction ..................................................................................... 3

2.2. A Review of Infinite Galois Extensions ................................................. 4

2.3. A Synopsis of Dedekind’s Article......................................................... 8

3. TRANSLATION ......................................................................................13

REFERENCES............................................................................................42

APPENDIX RELATED PROOFS ...................................................................43

BIOGRAPHY OF THE AUTHOR ..................................................................78

iii

LIST OF TABLES

Table 3.1. German expressions and terms used by Dedekind ................................41

Table A.1. Fixed field-closed subgroup correspondence. ......................................75

iv

LIST OF FIGURES

Figure A.1. Sublattice generated by He and Gpm ................................................46

Figure A.2. Corresponding sublattice of K = Q(⇣pn) (over Q)..............................46

Figure A.3. Subgroup lattice of Gal(Q(⇣2n/Q)) ................................................48

Figure A.4. Corresponding formal lattice of subfields of K = Q(⇣2n) (over Q) ........49

Figure A.5. Lattice of subfields of K = Q(⇣2n) (over Q)......................................50

Figure A.6. Intermediate field extension tower in ⌦/k .........................................55

Figure A.7. Composite lattice over a Galois & an arbitrary extension of k ...............58

Figure A.8. Closed subgroup (left) and corresponding field extension (right)

sublattices for Q(⇣p1) and etc., for fixed e|�(p) .................................74

v

CHAPTER 1

INTRODUCTION

In the 1901 paper “Uber die Permutationen des Korpers aller algebraischen

Zahlen” (“On the Permutations of the Field of All Algebraic Numbers”), Richard

Dedekind demonstrates the failure of the Fundamental Theorem of Galois Theory for

infinite degree algebraic extensions of the rational numbers Q. Dedekind’s original

work concerning infinite Galois Theory is in effect the genesis of a field whose begin-

ning is generally credited to Wolfgang Krull [7], some quarter century later. Dedekind

provides a coherent exploration of fundamental characteristics of infinite Galois exten-

sions (within an algebraic closure of Q). The supplement from Dedekind’s Nachlass,

published with the paper in the Collected Works, suggests that Dedekind in fact devised

a natural characterization of the closed subgroups of the Galois group, in Krull’s sense,

without the modern devices (e.g., general topology, general field theory, general set

theory with the Axiom of Choice) Krull had at his disposal.

Dedekind establishes the failure of the Fundamental Theorem for the case of

infinite Galois extensions over Q by showing that, for any infinite degree p-power cyclo-

tomic extension (for fixed odd prime p), there is a proper subgroup of its Galois group

that has the same fixed field as the full group.1 Cyclotomic extensions had been thor-

oughly studied before the 1890s, making construction of the counterexample as straight-

forward as possible. But in fact, until the mid-1890s, the entire subject of infinite-degree

extensions for Dedekind “hat bisher ein Noli me tangere gegolten [had hitherto been

considered a Noli me tangere (Touch me never) ].”2 It is perhaps then unsurprising that1Dedekind’s finding for a field of characteristic zero is found to hold for general characteristic, e.g.,

as when J. Neukirch shows that both the proper subgroup of the Galois group of the algebraic closureof F

p

generated by the Froebenius automorphism has precisely Fp

for its fixed field. The limitation toalgebraic subfields of C is a practical matter for Dedekind: The paper seeks to demonstrate the failure ofthe Fundamental Theorem of Galois Theory when the degree of the extension is nonfinite by constructinga single example.

2Letter to Frobenius, April 18, 1897, quoted in [3], V. II.

1

Dedekind sought a concise, efficient proof advancing the understanding of field theory

and its connections to Galois theory for the case of infinite degree extensions.

In this work, a modern presentation of Dedekind’s proof is first provided.

Our elementary description of his proof uses relevant contemporary mathematical

machinery. Thereafter an overview of the paper itself is given, connecting modern with

Dedekind’s concepts and terminology. Finally, an annotated translation of Dedkind’s

article is presented, as there is presently no English translation of the article available.

Appendix A provides proofs of results mentioned in the modern exposition.

2

CHAPTER 2

MODERN EXPOSITION OF RESULTS AND SYNOPSIS OF ARTICLE

2.1 Introduction

B. M. Kiernan, in “The Development of Galois Theory from Lagrange to Artin”

[6], describes the evolution of Galois theory through 1950 as having three stages. Its

initial phase was “computational,” developing the theory of equations. (This phase

includes work from Lagrange to Galois himself.) The second phase was “one of abstrac-

tion” involving the development of group theory and field theory. (Cauchy, Serret, and

even C. Jordan may be included in this phase.) The third phase was “one of generaliza-

tion, even universalization,” epitomized for Kiernan by E. Artin’s seminal work, Galois

Theory [1] establishing the theory as the study of the Galois groups of field extensions.

Further significant generalization and universalization of the theory was provided by W.

Krull [7], who extended the central theorem of the one-to-one correspondence between

the intermediate fields of a finite Galois extension and the subgroups of the Galois group

to the case of infinite Galois extensions. Krull did so by imposing a topology on the

group and deriving a one-to-one correspondence between the intermediate fields of the

Galois extension and the closed subgroups of the Galois group.

Bridging all three phases and contributing fundamentally to Artin’s and Krull’s

formulations is R. Dedekind. Dedekind was apparently the first mathematician to offer

lectures in algebra that covered Galois theory. Dedekind’s Gottingen lectures on higher

algebra given the winter semester 1857-1858 (see [12] for a published copy of the

lectures) introduced Galois’ concepts and results to German-speaking mathematicians,

including Henrich Weber. In the forward to his Lehrbuch der Algebra, Vol. 1, [14],

Weber mentions his appreciation of Dedekind’s 1857-1858 lectures, particularly his

inclusion of Galois theory. Dedekind’s subsequent consideration of Galois’ work was,

as with his non-constructivist approach to mathematics, with an eye toward abstraction

3

and well-founding. His reformulation of finite Galois theory, as a study of vector spaces

over subfields of the complex numbers and groups of isomorphisms (“permutations”) is

precisely surveyed in his Eleventh Supplement to Dirichlet’s Vorlesungenuber Zahlen-

theorie, [4]; cf. § 160–§ 166. This work provides one bridge between the second and

third phases of Galois theory. Indeed, comparison of Dedekind’s with Artin’s formu-

lations makes clear how closely Artin’s generalization and universalization of Galois

theory is indebted to Dedekind’s approach, extended to fields of arbitrary characteristic.

Dedekind built a second bridge to the modern theory of Galois theory in the 1890s

through his consideration of infinite-degree Galois extensions. His principal result

in this area, “Uber die Permutationen des Korpers Aller Algebraischen Zahlen” (see

[3], Vol. II, Article XXXI), provided the kernels of insight Wolfgang Krull needed to

successfully develop infinite Galois theory in the 1920s. Similar to Artin’s development,

Krull’s main ideas follow quite closely those presented by Dedekind, over a quarter of

a century earlier. One is reminded of Emmy Noether’s observation, “Es steht schon bei

Dedekind” (“It is already in Dedekind”).

The purpose of this work is to describe this second bridge, Dedekind’s investi-

gation into infinite Galois theory, and give a translation of the work. First we review

general Galois extensions, finite and infinite. We then give a synopsis of Dedekind’s

paper, after which we present our translation.

2.2 A Review of Infinite Galois Extensions

We first present a brief review of (finite and infinite) Galois theory from a modern

standpoint. To this end, let K/k be a Galois extension of fields, i.e., a normal separable

extension (not necessarily finite), within a fixed separable algebraic closure k, which

exists by the well-ordering principle.1 Let G = Gal(K/k) be its Galois group. Note we1See the article [13] of E. Steinitz. This work, published in 1910, is a rather modern axiomatic treat-

ment of field extensions (excluding Galois theory). The most popular types of fields at the turn of thetwentieth century were number fields and algebraic function fields. But this changed with Hensel’s devel-opment of p-adic numbers. As Steinitz mentions, this creation of Hensel motivated Steinitz to give an

4

have Gal(k/k) as a special case. Endow the Galois group G with the Krull topology,

by which we decree that a fundamental system of neighborhoods of the identity, idK ,

be the set of subgroups of the form Gal(K/E), where E is finite Galois over k and an

intermediate field k ✓ E ✓ K. The group G is a topological group under the Krull

topology. Notice that when G is finite, Gal(K/K) = {idK} is an open set and this

implies that G is discrete with respect to the Krull topology. (Cf. Appendix Section A.3.)

With this machinery we can state the fundamental theorem of Galois theory.

Theorem [Galois Correspondence] Let K/k be a Galois extension. Then the mapping

F 7! Gal(K/F )

gives a one-to-one correspondence between all intermediate fields F , i.e., k ✓ F ✓ K,

and all closed subgroups of Gal(K/k).

This theorem subsumes the fundamental theorem of finite Galois theory which

determines a one-to-one correspondence between all intermediate extensions and all

subgroups of G, since in the finite case the group G has the discrete topology, so in

particular all subgroups are closed. (Cf. Appendix Section A.4.)

The Krull topological group can also be described as a profinite group. Consider

the Cartesian productQ

=

Q

E Gal(E/k) of all finite Galois extensions of k contained

in K. Endowing the finite groups Gal(E/k) with the discrete topology, the Cartesian

product becomes a topological group with respect to the product topology (since the

product topology is defined as the coarsest topology on the product such that projections

onto components, (. . . , �E, . . . ) 7! �E, for all E as above, are continuous). Forming

axiomatic treatment of general field theory, including for the first time a study of extensions of fields ofnon-zero characteristic. This work includes a proof of the existence of algebraically closed fields usingthe well-ordering principle of Zermelo, based on ideas of G. Cantor.

5

the inverse (projective) limit

lim

�

E

Gal(E/k) =n

(. . . , �E, . . . , �F , . . . ) 2Y

: �F |E = �E,whenever E ✓ Fo

,

where �|E denotes the restriction of the map � to E, we observe that it is a closed

subgroup of the productQ

when endowed with the subspace topology.

The inverse limit is thus compact by Tychonov’s theorem and also totally discon-

nected. Indeed we find:

Theorem The mapping

Gal(K/k) ! lim

�

E

Gal(E/k)

defined by

� 7! (. . . , �|E, . . . ),

is a topological isomorphism between the two groups.

(Cf. Appendix Section A.5.)

We next consider, from a modern point of view, two examples Dedekind investi-

gated. The first example Dedekind investigates is Gal(Q/k) for any subfield k of Q, the

algebraic closure of Q in C, the field of complex numbers. Gal(Q/k) is the group of all

automorphisms of Q fixing k elementwise. By the surjectivity of the map in the previ-

ous theorem, any automorphism of a finite Galois extension E/k extends to an element

of Gal(Q/k). More generally, if F/k is any finite extension, then any isomorphism of

F into C fixing k extends to an element of Gal(Q/k), by first extending to the normal

closure of F and then repeating the previous argument.

Dedekind establishes that Gal(Q/k) is a group and that any isomorphic embed-

ding of k into C extends to at least one automorphism of Gal(Q/k), infinitely many

when [Q : k] is infinite. He also shows for this case that the mapping in the theo-

rem [Galois Correspondence] above is indeed injective. Initially however, as evidenced

6

in surviving manuscripts of the 1901 article, Dedekind also thought this mapping was

surjective when all subgroups are considered. However, he discovered a counterexample

by considering the sorts of infinite-degree cyclotomic fields to be described next.

The second example involves the infinite cyclotomic field Q(⇣p1) =

S

n2N Q(⇣pn)

where p is a prime and ⇣m = cos(2⇡/m) + i sin(2⇡/m) is a primitive mth root of unity.

If � 2 Gal(Q(⇣p1)/Q), then � is uniquely determined by its action on all the ⇣pn ,

for n 2 N. But Gal(Q(⇣pn)/Q) ' (Z/pnZ)⇥ where the automorphisms are given by

⇣pn 7! ⇣cnpn . Thus �⇣pn = ⇣cnpn for some cn 2 (Z/pnZ)⇥ and by compatibility cn+1 ⌘ cn

(mod pn). By taking projective limits, this gives us the following chain of topological

group isomorphisms:

Z⇥p ' lim

�

n

(Z/pnZ)⇥ ' lim

�

n

Gal(Q(⇣pn)/Q) ' Gal(Q(⇣p1)/Q).

(Cf. Appendix Section A.6.) Now the direct sum decomposition of (Z/pnqZ)⇥ for

n � 0, namely

(Z/pnqZ)⇥ ' (Z/qZ)⇥ ⇥ (1 + qZ)/(1 + qpnZ),

where q = p if p is odd and q = 4 if p = 2, induces, by taking projective limits, a

topological group isomorphism

Z⇥p ' (Z/qZ)⇥ ⇥ (1 + qZp).

The closed subgroups are readily determined to be

He ⇥Gm, for all e|'(q) and m 2 N [ {0,1},

where He is the subgroup of (Z/qZ)⇥ of index e and Gm = 1+qpmZp, with G1 = {1}.

7

(Cf. Appendix Section A.7.)

On the other hand, we can also list all the subfields of Q(⇣p1). Namely, let

Bp1q be the fixed field of the subgroup H1 = (Z/qZ)⇥ in Q(⇣p1) and furthermore, let

Bpmq = Bp1q \ Q(⇣pmq). Notice, too, that Q(⇣pmq) is the fixed field of Gm for all m

as above. Finally let Qe be the (unique) field in Q(⇣q) of degree e over Q. Then the

composite fields

F = Qe · Bpmq, for all e|'(q) and m 2 N [ {0,1},

exhaust all subfields of Q(⇣p1). Moreover the Galois correspondence is given by

F 7! Gal(Q(⇣p1)/F ) ' He ⇥Gm.

(Cf. Appendix Section A.8.)

As we will see, Dedekind determines this correspondence between the subfields

of Q(⇣p1) and this family of subgroups, when p is odd. Since there are other subgroups

of this infinite Galois group, the fundamental theorem in general fails to extend to the

infinite case if all subgroups are considered, as Dedekind proves.

2.3 A Synopsis of Dedekind’s Article

Dedekind’s article, as it appears in his collected works, contains six sections and

a short supplement based on material from his Nachlaß.

The first two sections offer a summary of finite degree algebraic field exten-

sions and Galois theory of finite extensions. In the first Dedekind defines a field (which

is always a subfield of C), subfields, intersection and composite fields (although with

different terminology), dependence and independence of finite sets of elements over a

field, algebraic elements over a field, finite degree extensions. The section ends with the

8

definition of an infinite degree extension of fields. No consideration of infinite cardi-

nal numbers is given here. Hence infinite means not finite. The second is devoted

to field isomorphisms, called “permutations” by Dedekind.2 He considers restrictions

and extensions of an isomorphism. The section ends with the isomorphism extension

theorem for finite algebraic extensions: If K/k is a finite degree extension and ' an

isomorphism on k, then there exists an extension of ' to K (in fact [K : k] of them).

In the third section, Dedekind considers infinite degree algebraic extensions

within C. He asks if it is possible to extend the non-identity automorphism on Q(

p

2 )

to an isomorphism of R into C. He then makes the conjecture that the only isomor-

phism of R into C is the identity map. This conjecture was shown to be false by

A. Ostrowski, [10], whose work is based very closely on that of Steinitz, [13]. Had

Dedekind considered only automorphisms on R, however, his conjecture would have

been correct. Dedekind then proceeds to show that any isomorphism on a subfield k

of Q extends to an isomorphism on Q, in fact infinitely many if Q/k is of infinite

degree. The proof uses an extension to Q of Cantor’s result that Q \ R is countable.

Let {an : n 2 N} = Q, where n 7! an is a bijection from N to Q, and let k be a

subfield of Q with [Q : k] = 1, (the case of finite degree is covered by the extension

theorem in the previous section). Now suppose ' is an isomorphism on k (into C). Let

n1 = min{n : an 62 k} and define k1 = k(an1). Continuing in the same way, Dedekind

obtains a chain of fields k1 ( k2 ( k3 ( . . . , with [kn = Q. Similarly, Dedekind

extends ' one step at a time. He defines '1 on k1 by requiring that '1(an1) = ar where

r is the minimal positive integer such that ar is a root of the irreducible polynomial of

an1 over k. Continuing, he obtains a chain of extensions ', '1, '2, . . . . Finally, he

shows that ['n is an isomorphism on Q.2Dedekind’s use of the term goes back at least as far as Lagrange, who considered permutations as

arrangements of the roots of a polynomial. Cauchy (1815) defined a permutation as a correspondence ofobjects, emphasizing the operation rather than the actual arrangements of the objects. It was with Galoisthat permutations evolved essentially into field isomorphisms.

9

Section 4 generalizes the main results of § 3 to arbitrary extensions K/k with K

a subfield of Q; namely, any isomorphism of k extends to one on K (infinitely many

when [K : k] = 1). This follows easily from § 3 by first extending to Q and then

restricting to K.

Section 5 finally introduces normal extensions and Galois groups. Dedekind

states the main result of the Galois theory of finite extensions: Given a finite normal

extension K/k, there is a one-to-one correspondence between the intermediate fields

and the subgroups of the Galois group G = Gal(K/k) of the extension. The argument

given uses the following theorem. If G is a finite subgroup of the group of automor-

phisms of a field K, then the fixed field, k, of G satisfies the equality |G| = [K : k].

Dedekind gives a proof of this result in [4]. Dedekind’s proof anticipates the modern

proofs of Artin, for example, in that Dedekind proves and then uses the independence

of the elements of G over K. This is all done for a comparison with the infinite degree

case handled in the last section.

In section 6, Dedekind starts by considering the group G = Gal(Q/k) for some

subfield k in Q. He defines a mapping from the intermediate fields F and subgroups of

G by the rule

F 7! Gal(Q/F ),

i.e., by mapping any intermediate field to its associated group. This gives, as Dedekind

shows, an injection from the fields to the subgroups. He then mentions that he tried in

vain to prove that any subgroup was the associated group for some intermediate field,

until he came up with a counterexample. As he also observes, in any counterexample

there would exist two distinct subgroups with the same fixed field.

To construct a counterexample, Dedekind considers the infinite cyclotomic

extension Q(⇣p1)/Q for fixed odd prime p (this is not a major restriction, cf. Appendix

A Section A.2), In effect, he shows that G = Gal(Q(⇣p1)/Q) is isomorphic to Z⇥p .

While Dedekind was constructing this counterexample, the p-adic numbers were being

10

formalized by K. Hensel. Still, Dedekind gives an equivalent characterization of the

p-adic units by, in essence, using the fact that they are characterized as an inverse limit

of the groups (Z/pnZ)⇥. From this, Dedekind is able to determine the torsion subgroup

of G. Since Z⇥p ' (Z/pZ)⇥ ⇥ (1 + pZp) ' (Z/pZ)⇥ ⇥ Zp , the last component as a

group under addition, which is torsion-free, we see (as Dedekind concludes) that the

torsion group is of order p � 1. (Recall that p is assumed to be an odd prime.) On the

other hand, he finds what in modern terms we would identify as a topological generator

� of G. Then as Dedekind notices, the fixed field of G and h�i, the cyclic subgroup

generated by � is, in both cases, Q.

But he then observes that h�i 6= G.

Though today we would satisfactorily conclude the matter with the observation

that |Z⇥p | = |R| whereas |h�i| = |N|, Dedekind argues without consideration of cardi-

nality. Instead, he notes that the torsion subgroup of G has order p � 1 > 1, while the

torsion subgroup of the infinite cyclic group h�i consists of the identity map only. This

counterexample establishes that the fundamental theorem of finite Galois theory does

not hold of necessity for infinite extensions.

Dedekind does not list all the intermediate fields and all corresponding fixed

groups. He mentions in the article that he could but would not do so in the paper. This is

the subject of the supplement. Dedekind also makes an imprecise remark after relating

the automorphisms in Gal(Q(⇣p1)/Q) to (in essence) Z⇥p . He says that Gal(Q(⇣p1)/Q)

forms, in a certain sense, a continuous manifold. He does not provide details to deter-

mine what he means.3 Krull observes that this remark is a fundamental idea that he

exploits in endowing general Galois groups with the appropriate topology, cf. [7].3In Stetigkeit und irrational Zahlen, see [3], Vol. III, Dedekind observes that R is in essence a topo-

logical ring. Hence it is perhaps not too much of a stretch to suspect that he has an analogous topologicalgroup structure in mind here.

11

In the supplement added to the article when his collected works were published,

Dedekind (or perhaps the editors) gives the list of all subfields and corresponding asso-

ciated groups of the extension Q(⇣p1)/Q (p again an odd prime). As we now know,

these subgroups are precisely the closed subgroups of (via an isomorphism) Z⇥p . In

retrospect these subgroups are apparent from the topological isomorphism of Z⇥p and

(Z/pZ)⇥ ⇥ (1 + pZp). Closed subgroups are then of the form

He ⇥ h1i and He ⇥ (1 + pmZp),

where He is a subgroup of (Z/pZ)⇥ of index e (so e is a divisor of p � 1) and m is

an positive integer. Subfields corresponding to these groups are then, respectively, the

composites

Qe · Bp1 and Qe · Bpm ,

where Qe is the unique subfield of Q(⇣p) of degree e over Q, Bp1 is the fixed field of

(Z/pZ)⇥ in Q(⇣p1), and Bpm = Bp1 \Q(⇣pm).

12

CHAPTER 3

TRANSLATION

1On the Permutations 2 of the Field of all Algebraic Numbers.

The purely algebraic presentation considered here has the goal of extending certain theo-

rems on finite fields to infinite fields.3 But in order to better appreciate the subject, it is

necessary to recall the meaning of the expressions in the title of this paper, as well as

some theorems related to them. One can find a detailed development of these concepts

and the proofs in the fourth edition (1894) of Dirichlet’s Vorlesungen uber Zahlentheo-

rie (Supplement XI), which I shall allude to by (D.) in the following.4 Here, in the first

two sections, I restrict myself to extracting from this work (with proofs omitted) only

that which is indispensable for our purposes.

1Note that two types of footnotes appear in this translation: Numeric footnote by the translators (suchas this footnote), and Dedekind’s original footnotes, which are marked here by one or more asterisksfollowed by a right parenthesis, followed by a numeral.

2As we keep to Dedekind’s terminology, a chart of Dedekind’s terms and contemporary equivalencesis given at the end of this translation. In particular, Dedekind’s use of “permutation” corresponds to thecontemporary notion of field isomorphism. An evolution of vocabulary of field and Galois theory seemsto have occurred between 1900 and the 1920’s, presumably with the rise of the abstract, structuralistapproach to conceptualizing mathematics. E.g., Baer and Hasse’s Appendix (1929) of their edited andexpanded reprint of Steinitz (1910) explicitly defines an automorphism as an isomorphism mapping froma field to itself, and Artin’s (1942) vocabulary agrees with contemporary usage.

3Here, for Dedekind, finite (infinite) fields mean finite- (infinite-) degree field extensions.4Throughout our annotative footnotes we will not distinguish between “D.” as Dirichlet or Dedekind,

since Suppliment XI is the only portion of the Vorlesungen to which Dedekind refers, and Dedekind wrotethe supplement.

13

§ 1.

Fields and Irreducible Systems.

A system5 A of real or complex numbers is called a field (D. § 160), if the sums,

differences, products, and quotients 6 of any two of these numbers belong to the same

system A. The smallest field R consists of all rational numbers, the largest field Z of all

complex numbers. A field A is called a divisor of a field B, and at the same time B is

called a multiple of A, when every number contained in A also belongs to the field B.7

The field R is a common divisor, the field Z a common multiple of all fields A. If B is a

multiple of A and a divisor of C, then A is a divisor of C. Every given system of fields

A, whether finite or infinite, has a uniquely determined8 greatest common divisor D;

this field consists of those numbers which are common to all these fields A, and every

divisor common to all these fields A is a divisor of D. The same system of fields has a

least common multiple M ; this field M is the greatest common divisor of all those fields

which (as, e.g., Z) are a common multiple of the fields A.9

A finite system T of m numbers t1, t2, . . . , tm is called reducible with respect to

the field A, if there exist m numbers a1, a2, . . . , am in A, not all vanishing, which satisfy

the condition

a1t1 + a2t2 + · · ·+ amtm = 0.

In the contrary case the system T is called irreducible over A (D. § 164).10

5Das System is colloquial, much as collection in English, and is to be distinguished from der Inbe-griff, which carries the sense of a totality of things, a collection of things such that . . ., which we willunderstand and translate as a set (of elements) such that. Dedekind does not use Cantor’s term, die Mengefor set, although Dedekind had followed Cantor’s development of Mengenlehre (set theory) well beforethe 1896 publication in Mathematische Annalen of Cantor’s “Beitrage zur Begrundung der transfinitenMengenlehre.”

6In D. § 160 it is explicitly noted that the denominator of a quotient cannot be 0.7Dedekind uses “divisor” for subfield and “multiple” for extension field. Cf. the table on the last page

of this chapter.8“...besitzt einen bestimmten...”9 I.e., D is the intersection of the set of fields A; M , their composite.

10In D. § 164, these terms are introduced in the same order. The text goes on to say that ”[a]ccordingto whether the form or latter case occurs, we will also say that the m numbers . . . are dependent on, or

14

A number t is called algebraic with respect to the field A, if there is a natural

number n, for which the n+ 1 powers

1, t, t2, . . . , tn�1, tn

form a reducible system over A; the smallest number n, for which this occurs, is called

the degree of t, and we say that t is an algebraic number of nth degree with respect to A.

Obviously such a number t is the root of an (irreducible) equation

tn + a1tn�1

+ · · ·+ an�1t+ an = 0,

whose coefficients a1, a2, . . . , an are numbers of the field A, whereas the n powers

1, t, t2, . . . , tn�1

form an irreducible system over A. One can easily see (D. § 164, IX) that the set U of

all numbers u of the form

u = x1tn�1

+ x2tn�2

+ · · ·+ xn�1t+ xn,

where x1, x2, . . . , xn�1, xn denote any arbitrarily chosen numbers in A, is again a field,

and indeed is a multiple of A.11 We denote this field U by A(t), and we say that it is

generated from A by adjoining t. Every n+1 numbers of this field A(t) form a reducible

system over A, whence every number u is algebraic with respect to A, and its degree is

no larger than n.12 If this degree = n, then A(u) and A(t) are identical.

independent of, each other (over A).” I.e., a reducible (resp. irreducible) set is linearly dependent (resp.independent) over A.

11So the set (Inbegriff ) is closed under inversion; i.e., A[t] = A(t).12See D. §164, VII.

15

A field B is said to be finite13 with respect to the field A and of degree n, if there

is in B an irreducible system over A consisting of n numbers, while any n+ 1 numbers

of the field B form a reducible system over A. We denote this degree n, which is always

a natural number, by the symbol (B,A).14 Then all the numbers in B are algebraic with

respect to A, and among these there are also (infinitely many) numbers t, whose degree

= n (D. § 165, VI),15 and the field A(t) generated from A by adjoining t is the least13It should be clear that Dedekind’s use of “finite field” is not synonymous with the contemporary

notions of “finite field” (Dedekind considers only subfields of C), “finite degree extension field” (seefootnote following), or even “finite dimensional vector space” (since, as fields, not all products of scalarsin A and vectors in B are necessarily closed with respect to B).

14Notice here that A need not be a subfield of B. For example, by Dedekind’s definition(Q(

p

3 ), Q(

p

2 )) = 2.15Perhaps Dedekind should have put this parenthesis at the end of the sentence, as D § 165, VI includes

that A(t) = M for each such t. It may be of interest here to note that Dedekind’s proof relies on resultsin D § 161 obtained prior to the introduction of algebraic elements or vector spaces (in D § 163). In § 161,Dedekind shows that if � = {�

i

}

n

i=1 is a set of n distinct isomorphisms, each with domain A (a field),then there exists in A infinitely many numbers which are n-valued with respect to �. To explain: In fullgenerality, Dedekind first considers a set � of n arbitrary field isomorphisms, restricts to their commondomain A (a subfield of C), then partitions the common domain, each partition containing elements whoseorbit under � is a fixed cardinality. Accordingly, an element ↵ of the common domain is said to be one-valued, two-valued, etc. with respect to � if the orbit of ↵ consists of one, two, etc. elements (cf. § 2below). As all fields under consideration are subfields of C, certainly all elements of the prime subfield Qare one-valued for every possible �. To prove D § 161, Dedekind must exhibit at least one such n-valuednumber. The nontrivial part of the proof by induction considers the case when, for some a 2 A, theimages a�

r

= ar

are distinct for r > 1 but, without loss of generality, a1 = a2. In detail, as �1 6= �2one can find some b 2 A such that b1 6= b2. Now for any x 2 Q surely y = ax + b 2 A and soy�

r

= yr

= ar

x+ br

, also yr

� ys

= (ar

� as

)x+(br

� bs

), (r, s 2 {1, 2, . . . , n}, r 6= s). In particular,for r = 1 and s = 2, since a1 = a2 and b1 6= b2, one finds y1 � y2 6= 0. For any other choice of r, s,ar

6= as

, so yr

= ys

if and only if b

s

�b

r

a

r

�a

s

2 Q. Then choosing x 2 Q � {

b

s

�b

r

a

r

�a

s

|r 6= 1, s 6= 2}) ensuresthat y is n-valued with respect to �. Dedekind further notes here a connection with the Vandermondedeterminant: As all y

r

� ys

with r < s are surely non-zero, so is their product

Q

1r<sn

(yr

� ys

) =

�

�

�

�

�

�

�

�

yn�11 yn�2

1 · · · y1 1

yn�12 yn�2

2 · · · y2 1

. . . . .yn�1n

yn�2n

· · · yn

1

�

�

�

�

�

�

�

�

non-zero. Now with respect to D § 165, VI, the proof proceeds by choosing arbitrary ✓ 2 M = AB andconsidering the set T of periodic elements ✓r, r 2 {0, 1, 2, . . . , n�1}; or, more precisely, considering thedeterminant (T ) = det (✓r⇡

s

), s 2 {1, 2, . . . , n}, which he defined while proving D § 165, IV (that thenecessary and sufficient condition for a set T ✓M to be linearly independent over A and form a basis forM is for (T ) 6= 0). The ⇡

s

are elements of ⇧, the set of all isomorphisms of M that restrict to the identityisomorphism on A. But now (T ) = det (✓r⇡

s

) = det ((✓⇡s

)

r

) is a Vandermonde determinant of theperiodic elements, so (T ) =

Q

1r<sn

(✓⇡r

�✓⇡s

). Hence the set T is (by D § 161) independent over Aif and only if ✓ is n-valued with respect to ⇧. Since B is a field of nth degree over A, every independentset of n numbers in B or M forms a basis for M over A (D § 164, VIII). Thereby every element of Mis algebraic with respect to A and of degree at most n. Thus every n-valued number ✓ and no other isof degree n. But again, from § 161, since consists of n distinct isomorphisms of field B, there must

16

common multiple M of the two fields A,B. Hence we have the statement

(B,A) = (M,A). (3.1)

If B is itself a multiple of A, so M = B, then in this case B is called a finite

multiple of A; if in addition the field C is a finite multiple of B, then C is also a finite

multiple of A, and the following statement holds (D. § 164, X)

(C,A) = (C,B)(B,A). (3.2)

exist infinitely many numbers ✓ that are n-valued in , hence (by extension) in ⇧. This proof yields aversion of the Theorem of the Primitive Element for finite degree algebraic extensions. (Dedekind willimmediately consider the question of the number of possible intermediate fields K given an algebraicextension field B of field A of degree [B : A] = n and show there can be but finitely many intermediatefields distinct from A and B. Interestingly, Dedekind does so by noting how K is uniquely determinedby some subgroup ⇧

0of ⇧ and the possible partitions of ⇧ by cosets of subgroups is finite.)

It might also be noted in passing that Dedekind was well aware that should the field M be normal overA, with ⇧ the automorphism group of M/A, the determinant (T ), if non-zero, gives a cyclic presentationof the elements ✓r whose value can be expressed in terms of the elements and group characters of the ⇡

s

.Cf. [5]. Lang [8] notes that Dedekind “expressed the theorem [of the independence of group characters]”in the case when “K is a finite normal extension of a field k, and when the characters are distinct automor-phisms [�

s

]... of K over k... [which he proved] by considering the determinant constructed from �i

!j

where !j

is a suitable set of elements of K, and proving...this determinant is not 0.” This is precisely theform of the determinant (T ).

Tangentially, Lang describes Dedekind’s proof in contrast to Artin’s proof [1], indicating that theform of the theorem in [1] and “its particularly elegant proof are due to Artin.” Lang’s emphasis onhis former teacher’s elegance does not account for the degree to which Artin’s elegance may have beenachieved using the template Dedekind provides in Supplement XI . By contrast, Kiernan [6] notes that“much of [ARTIN’s] work was prefigured in the presentations of DEDEKIND and WEBER” (144) andthat Artin “took up the concept stated implicitly by GALOIS and announced, unheard, by DEDEKIND andWEBER...[namely, that the] theory is concerned with the relation between field extensions and their groupsof automorphisms,” (emphasis added) and that “Armed with DEDEKIND’s view of a field extension as avector space over the ground field...ARTIN began the discussion with the splitting field of the polynomial”(emphasis in original). Still, Kiernan does not seem to appreciate the apparent parallels between Artin andDedekind’s presentations (after making appropriate adjustments for the great conceptual developments inmathematics between the 1890s and the 1940s.) Not only are the large-scale outlines the same (introduc-tion of fields, followed by vector spaces, followed by algebraic field extensions and iso/automorphsimson such), but the structure of at least five of Artin’s key proofs parallel Dedekind’s (Theorem 6 parallels§ 165 X, Theorem 14 parallels § 166 I, Theorem 26 parallels § 165 VI, converse, Theorem 27 parallels§ 165 VI). Even an unmotivated detail such as Artin’s definition of a fixed point of a field (with respectto some set of isomorphisms) aligns with Dedekind’s definition of a one-valued number (with respect tosome set of isomorphisms). By attentively summarizing sections 160-166 of the Supplement, Dean [2]provides a substantial argument that Artin’s version of the Fundamental Theorem of Galois Theory is infact Dedekind’s version, excepting the generalization to fields of arbitrary character. It appears however,at present, that the literature contains no detailed analysis of structural and conceptual similarities anddifferences between Dedekind’s and Artin’s presentations of Galois Theory by which to evaluate thevalidity of apparent parallels.

17

That D is a divisor of field M can be completely characterized by (D,M) = 1.

But if B is not finite with respect to A—so that there always exists in B, however

large a natural number m as we choose, m numbers in B which form an irreducible

system over A—then we shall set (B,A) = 1.*)16 Whereby we see that the two state-

ments (3.1) and (3.2) hold in general, the latter of course subject again to the earlier

assumption that B is a multiple of A and divisor of C.17

§ 2.

Permutations of a Field.

A function ' of a field A, by which every number a contained in A is mapped to

a corresponding number a', is called a permutation of A, if the four laws

(u+ v)' = u'+ v', (u� v)' = u'� v',

(uv)' = (u')(v'),⇣u

v

⌘

' =

u'

v'

are obeyed, where u, v are arbitrary numbers in A (D. § 161); we also say that the permu-

tation ' is defined on A and call for brevity the latter the field of ', in order to emphasize

the fact that the mapping ' may not be applied to any number not in A.18 If furthermore

T is a part of A, i.e., a system of numbers19 t, which are all contained in A,20 then we16*) Compare the conclusion with D. § 164, where for this case (B,A) = 0 was given. This is less

advantagous in the present work.17Dedekind takes no account here of the cardinality of these sets, which provides another avenue to

proving the failure of the Fundamental Theorem of Galois Theory in the case of an infinite degree Galoisextension of Q. Dedekind seems to maintain a conceptual distinction between an “infinite set” and an“infinite number.” Yet, insofar as Dedekind has the dimension of a vector space over a base field in mind, itis not clear why he would not have considered the degree of an extension in determining the equipollencyof sets, in which case, a nonfinite degree extension would be subject to cardinality considerations.

18The difficulty of extensions is foreshadowed: How to enlarge the domain is nontrivial.19“...ein System von Zahlen...”20I.e., T is a subset of A.

18

denote by T' the set of all images21 t' of these numbers t. The number t' is said to be

conjugate to t.

From this definition it follows easily that the number system A' is again a field,

and that every two distinct numbers of A map under ' to two distinct numbers in the

conjugate field A'.22 For this reason, the permutation ' can be inverted, and if one

denotes by '�1 the mapping of the field A', by which every number a' contained in

A' is mapped to a, then it is clear that the inverse '�1 is a permutation of A', and that

(A')'�1= A.

Every field A possesses at least one permutation, namely the so-called identity

permutation, by which each of its numbers is mapped to itself; we shall denote it in what

follows by A0.23 If ' is an arbitrary permutation on A, and r a rational number, which is

therefore also contained in A, then it follows that r' = r, whence the field R of rational

numbers has only a single [permutation], the identity permutation R0.

If the field A is a divisor of the field B, then for every permutation of B,

there is a corresponding permutation ' of A, which is defined by a' = a for each

number a of the field A; whence it follows at once that the field A' = A is therefore

a divisor of B . This permutation ' is called the divisor with respect to A of , and

at the same time, is called a multiple of ' with respect to B (D. § 163).24 In the case

that A = B, obviously ' = ; but if A is distinct from B, thus a so-called proper

divisor of B, then ' is substantively different from , since the domains of definition

of the two permutations are distinct. The sole permutation R0 of R, the field of rational

numbers, is the common divisor of all field permutations, and every divisor of an identity

permutation is likewise an identity permutation. If the permutation ' of the field A is21”...den Inbegriff aller Bilder...”22See D. §161, p. 458.23It is curious that Dedekind uses this symbol, for all other maps are denoted by Greek letters; more-

over, this symbol could be confused as representing a field, consistent with his notation later in this article.In (D.), the identity map is not assigned a particular symbol, though it is repeatedly represented by thegeneric symbol '.

24Hence a divisor, resp. multiple is a restriction, resp. extension map.

19

a divisor of the permutation , and further if the latter is a divisor of a permutation �,

then ' is the divisor of � with respect to A.

From the infinite collection of theorems25 related to these concepts, we wish to

single out here only two which are particularly important; in order to be able to state

them conveniently, we start with the following discussion (D. § 161). If P is a (finite or

infinite) system of permutations of arbitrary fields B, then a number t contained in

the greatest common divisor of these fields B is mapped by every permutation to a

corresponding number t , and it is called n-valued by P, if n is the number of distinct

values which appear among the numbers t . Clearly every rational number is single

valued by P. From this our first theorem, which is easily proved (D. § 163), is given as:

I. If P is a system of field permutations , then the totality of all the numbers

single valued by P forms a field A; the permutations all have one and the same divisor

' with respect to A, and every common divisor of all the permutations is a divisor of

this permutation '.

For brevity we call this field A, which is completely determined by the system

P, the field of P, and its permutation ' will be called the greatest common divisor of

the permutations or in brief the residue of P. If the system P consists of only a single

permutation , then clearly A is in the earlier sense the field of , and ' = .

Whereas the existence of divisors of a given field permutation is immediately

seen, the reverse question is much deeper; it can be answered at least in part by the

following second theorem (D. § 165, III):

II. If the field B is a finite multiple of the field A,26 and ' is a permutation of

A, then the degree (B,A) is also the number of distinct permutations of B, which

are multiples of '; moreover A is the field and ' the residue of the system P of these

permutations .25”...Menge von Satzen...”26B/A is not necessarily normal.

20

The most well-known example of this theorem arises by considering the field

Z of all complex numbers and the field X of all real numbers. Obviously, Z = X(i),

where i is a root of the quadratic equation i2 + 1 = 0; the two numbers 1, i form an

irreducible system over X , and every number in Z is uniquely representable in the form

x1 + i x2, where x1, x2 are contained in X , and consequently (Z,X) = 2. Now if '

denotes the identity permutation of X , then there are actually two and only two distinct

permutations on Z, which are multiples of '; one is the identity permutation of Z,

while the other is defined by (x1 + i x2) = x1 � i x2.

§ 3.

Permutations of the Field of all Algebraic Numbers.

The most recently emphasized Fundamental Theorem II assumes that the field B

is a finite multiple of the field A; if we omit this assumption, then it seems to me that the

answer to the question of whether every permutation ' of A has at least one multiple

with respect to B is of the greatest difficulty to determine. Let us consider, for example,

the real quadratic field A = R(

p

2 ), which is formed from the rational field R by

adjoiningp

2. Then A has a permutation ', which is not the identity permutation, by

whichp

2 is mapped to�p

2 , and since A is a divisor of the field X of all real numbers,

the following question thus arises: Does there exist a multiple of ' with respect to X? I

do not know, but nevertheless I believe that the answer is “No”. The numbers in the real

field X seem to me to be so intimately connected by continuity that I suspect that there

cannot exist any permutation of X other than the identity. From this it would follow that

the field Z possesses only the two permutations given at the end of § 2.27 After several

futile attempts to come up with a proof, I gave up this research; it would thus make me27This conjecture turns out to be false, cf. Ostrowski [10]. However, if Dedekind had considered only

automorphisms on R, then he would have been correct in that the only automorphism on R is the identitymap.

21

all the more happy, if another mathematician wanted to inform me of a definitive answer

to this question.

But the same question can be completely answered if we confine ourselves to the

discontinuous domain H of all algebraic numbers. By an algebraic number, we simply

mean here any number t which is algebraic with respect to the rational field R, thus a

root of an equation

tn + a1tn�1

+ a2tn�2

+ · · ·+ an�1t+ an = 0

with rational coefficients a1, a2, . . . , an. The set H of all these numbers t is a field, as is

well known, and by an algebraic field we merely mean any divisor of H; clearly H is

not a finite multiple of R, thus (H,R) = 1. We mention further that every conjugate

number t (§ 2) of an algebraic number t is likewise algebraic; for because the rational

coefficients a1, a2, . . . , an are fixed by every permutation , t must satisfy the same

equation as the root t. From this we proceed to the proof of the following existence

theorem:

III. If ' is a permutation of an algebraic field A, then the field H of all algebraic

numbers possesses at least one permutation ! which is a multiple of '.

If H is a finite multiple of A, then our theorem is an immediate consequence

of the main theorem II mentioned above (in § 2);28 hence we confine ourselves in the

following to the contrary case that (H,A) = 1, while (A,R) can be finite or infinite.

The proof is then based principally on an important property of the field H , first empha-

sized by G. Cantor*)29, and which is that all the numbers of the field H can be ordered28Dedekind in his handwritten manuscript explicitly gives A = H \R.29*) Uber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. (Crelles Journal, Bd. 77).

I had also discovered this result extended to the field H , but I had my doubts about its utility, until Ithought better of it due to the beautiful proof of the existence of transcendental numbers, which Cantorintroduced in § 2 of his paper.

22

in a simple infinite sequence

h1, h2, h3, . . . , hr, hr+1, . . . (h)

in such a way that every natural number r corresponds to a unique algebraic number

hr, and conversely that every algebraic number t corresponds to one (and only one)

natural number r, for which hr = t. Such an ordering (a mapping of the field H by the

sequence of natural numbers r) can be accomplished in infinitely many different ways;

for our proof we consider a fixed ordering (h), and we call the natural number r the

index of the algebraic number hr.

Now since we have assumed (H,A) =1, A is a proper divisor of H , i.e., there

exist in H , thus in the sequence (h), numbers which are not contained in A; among

all of these numbers there is a unique number t = hr, which has the smallest index r,

and we call this number r the index of the field A. If r > 1, then the r � 1 numbers

h1, h2, . . . , hr�1 preceding the number t all lie in A. Now since the number t is algebraic

with respect to A, the field

A1 = A(t),

obtained by adjoining t to A (by § 1) is a finite multiple of A and also a divisor of H . For

brevity we shall call this field A1, which is uniquely determined by A and our ordering

(h), the succeeding multiple of A. But the field H cannot be a finite multiple of A1, for

otherwise [by (3.2) in § 1] H would be a finite multiple of A, whence (H,A1) = 1.

One can hence apply the same reasoning to A1 as with A, and so by continuing in the

same manner we obtain from A an infinite chain (A) of fields

A, A1, A2, . . . , As, As+1, . . . , (A)

23

in which each successive term As+1 is the succeeding multiple of the previous As, while

at the same time they are all divisors of H . Furthermore, since the field A1 = A(t)

contains the new number t = hr along with the numbers h1, h2, . . . , hr�1 which are

already in A, the index of A1 must be� r+1, and by repeating this argument, it follows

that the field As surely contains all those numbers in the sequence (h) whose index is

< r + s. Now since every algebraic number has a uniquely determined finite index,

so that if one or more such [numbers] u, v, . . . be given, then we can always choose

a natural number s sufficiently large, so that all these numbers are in the field As, and

thence also contained in all the successive fields As+1, As+2, . . . .

Now let us assume that ' is an arbitrary permutation of the field A. Since the

number t = hr is not contained in A, and so the finite degree (A1, A) � 2, there

always exist, by Fundamental Theorem II (in § 2), [which is] valid for finite multi-

ples, several distinct permutations of the field A1 = A(t) which are multiples of

', and every permutation is completely determined by the conjugate number t to

which t is mapped. For our proof it would be completely immaterial whichever of these

(A1, A) permutations we wished to select; but in order to obtain a well-defined rule,

we proceed as follows. Since the numbers t (as is mentioned above) are likewise

algebraic, and are thus contained in the sequence (h) and moreover are all distinct, we

therefore decree to choose that permutation for which the index of t is as small as

possible. This permutation of A1, which is completely determined by ' and the order-

ing (h), we denote by '1 and call it the succeeding multiple of '. Clearly one can

now proceed with this permutation '1 of the field A1, just as with the permutation ' of

the field A, and by continuing in this way, we obtain from the given permutation ' an

infinite chain

', '1, '2, . . . ,'s, 's+1, . . . , (')

in which the general 's is a permutation of As, and 's+1 the succeeding multiple of 's.

24

Once the two chains (A) and (') of the fields As and their permutations 's have

been formed, the proof of our theorem III proceeds very easily. We define a mapping

! on H in the following manner. If u is any algebraic number, then there exists, by an

earlier observation, a field As in the chain (A), in which the number u is contained, and

if n denotes the smallest number s for which this holds, then we decree that u under !

be the image

u! = u'n.

By this, the mapping ! on H is completely determined, and we now wish to prove that it

is a permutation of H , and moreover a multiple of '. First we observe that the number u

of the field An is also in all the successive fields An+1, An+2, . . . of the chain (A), thence

in general contained in As, when s � n, and since at the same time 's is a multiple of

'n, it follows from the definition of ! that

u! = u's.

If now v is likewise an algebraic number, then one can choose s so large that the two

numbers u, v and consequently also their sum, difference, product, and quotient belong

to the same field As; whence it follows, as just observed, that

u! = u's, v! = v's,

(u+ v)! = (u+ v)'s, (u� v)! = (u� v)'s,

(uv)! = (uv)'s,⇣u

v

⌘

! =

⇣u

v

⌘

's.

Now since 's is a permutation of the field As, and thus obeys the laws given in § 2, it

follows immediately that the map ! of the field H obeys the same laws, therefore is a

permutation of H . Furthermore if a denotes any number in the field A, then it follows

from the definition of ! that a! = a', and therefore ! is a multiple of ', q.e.d.

25

§ 4.

Generalization.

In the following observations we now wish to extend and generalize theorem III

which was just proved; as always H denotes the field of all algebraic numbers. First

one easily sees that the theorem remains valid if the field H is replaced by any algebraic

field B which is a multiple of A. Namely, if ' is again a permutation of A, then as we

now know, there exists at least one permutation ! of H which is a multiple of '. Now if

denotes the divisor of ! with respect to B, then by an earlier observation (§ 2), ' is at

the same time the divisor of with respect to A. Thus we have the following theorem:

IV. If the field A is a divisor of the algebraic field B, then every permutation of

A has at least one multiple with respect to B.

When B is a finite multiple of A, then this is clearly only a special case of

theorem II (in § 2), which at the same time yields the stronger condition, that the degree

(B,A) is the exact number of all the distinct permutations . We can now also easily

prove, in the contrary case when B is not a finite multiple of A, that the number of

permutations of B, which are multiples of the same permutation ' of A, is infinitely

large, therefore again = (B,A), if we retain the meaning of the symbols given at the

end of § 1.30 To this end, consider any field A0, which (as, e.g., A itself) is a finite

multiple of A and at the same time a divisor of B. Since every such field A0 differs

from B, and so is a proper divisor of B, there certainly exist in B numbers t which are

not contained in A0, and consequently there is a field A00= A0

(t) obtained from A0 by

adjoining one such algebraic t. The field A00 is a finite multiple of A0, thus also of A,

and is again at the same time a divisor of B. Furthermore, since (A00, A0) � 2, and

so (A00, A) = (A00, A0)(A0, A) � 2(A0, A), it is clear that, if m denotes any arbitrarily

large natural number, then among all the fields A0 there is also such a one for which30This extension is valid since Dedekind defines infinite as not finite. However, the result is false if we

consider cardinalities. For example, [Q : Q] = |N|, whereas |Gal(Q/Q)| = |R|.

26

(A0, A) � m. By theorem II (in § 2) such a field A0 certainly possesses at least m

distinct permutations

'01, '

02, . . . ,'

0m,

which are multiples of the given permutation ' of A, and since A0 is a divisor of B, each

of these m permutations '0 has by theorem IV at least one multiple with respect to B;

the resulting m permutations

1, 2, . . . , m

are consequently also multiples of ', and they are all distinct, because each permutation

of B has just one, uniquely determined divisor '0 with respect to A0. Since m can be

taken arbitrarily large, it follows that the number of all permutations of B, which are

multiples of the same permutation ' of A, is infinitely large, therefore = (B,A), q.e.d.

Lastly, under the same assumption, we want to show that the last part of theorem

II (in § 2) also remains valid. The system P of all permutations of B, which are

multiples of the permutation ' of A, possesses (by theorem I in § 2) a greatest common

divisor �, and since ' is a divisor of all permutations , and so a divisor of �, it follows

that the field C of this permutation � is a multiple of A and at the same time a divisor of

B. Now assume that C differs from A, and so (C,A) � 2, then C possesses, as was just

proved, at least one permutation �0 distinct from � and which is likewise a multiple of

'. Furthermore, since C is a divisor of B, it follows that B has at least one permutation

0 which is a multiple of �0, and thus also a multiple of '; consequently, it belongs to

the system P. Whence �, being a residue of P, must be a divisor of 0, too. Therefore

0 would possess two different divisors �,�0 with respect to the same field C, which is

impossible. Our assumption above, that the fields A,C are different, is thus not possible,

and whence it clearly follows that � = '. From this we can extend theorem IV above in

the following way:

27

V. If the field A is a divisor of the algebraic field B, and ' a permutation of A,

then the degree (B,A), be it finite or infinite, is the number of all distinct permutations

of B which are multiples of '; also, A is the field and ' the residue of the system P

of these permutations .

§ 5.

Groups of Permutations.

It follows immediately from the preceding that the field H of all algebraic

numbers has infinitely many different permutations !. Now since every number t

contained in H is mapped by a permutation ! again to an algebraic number t!, it

follows that the conjugate field H! is clearly a divisor of H; but we can easily show

that always H! = H . For consider again any equation with rational coefficients, which

has as a root a given algebraic number t, and denote by t1, t2, . . . , tm all the m distinct

roots of this equation, which thus also belong to the field H . These numbers are mapped

(as mentioned in § 2) by ! to as many likewise distinct numbers t1!, t2!, . . . , tm!; but

since the latter satisfy the same equation, one of them must coincide with the given

number t. Therefore, every number t of the field H also belongs to H!, from which the

above claim that H! = H clearly follows. We signify this property of the field H , by

which all its permutations map it into31 itself, by calling it a normal field (cf. D. § 166).

An immediate consequence, or rather merely another formulation, of this property is

that the inverse !�1 of a permutation ! of H is also a permutation of H (§ 2).

Now it is time to recall a concept from the general theory of field permutations,

namely their composition (D. § 162). For this we restrict ourselves for brevity to the

following special case *).32 Let M be an arbitrary field, and E the set of permutations "31We would say onto, as normal should always be within an algebraic closure. Perhaps Dedekind is

considering transcendental elements.32*) By the way, I wish to observe here, that one can define in complete generality the composition

' when ', are permutations of two arbitrary fields A,B ; there exists a unique divisor A0 of A which

28

of M , by which M is mapped into M , thus satisfying the condition M" = M .33 34 There

is always at least one such [permutation], namely the identity permutation M0 of M , and

every inverse "�1 is likewise in E. Now every two (equal or distinct) such permutations

', generate a resultant ' , which is defined for each number x contained in M by

x(' ) = (x')

and is likewise a permutation of M in E. From this the two statements

(' )�1= �1'�1, (' )� = '( �)

follow, where � is also any permutation in E, and the resultant ''�1 is the identity

permutation M0. A system A of permutations ↵ contained in E is called a group, if

1. the resultant of any two permutations ↵, and 2. all inverses ↵�1 belong to the same

system A, and it is called finite or infinite according as the number of permutations ↵ is

finite or infinite; in the first case, one can easily conclude that the condition 2. is already

a necessary consequence of condition 1. The set E is itself a group, and also the identity

permutation M0 alone forms a group, which is contained in every group A. For finite

groups the following fundamental theorem holds (D. § 166, I):

VI. If a finite group A consists of n permutations of the field M , and if A is

the field of A,35 then (M,A) = n, M a finite multiple of A, and the residue of A is

is mapped by ' to the greatest common divisor of A' and B, and the composition ' is defined asthe permutation of A0 given by x(' ) = (x') , where x is any arbitrary number in A0. The twostatements (' )�1

= �1'�1, and (' )� = '( �) remain valid, whereas other statements requirecertain modifications.

33I.e., an automorphism, but Dedekind does not conceptually isolate such permutations.34Cf. footnote 28. To conclude that the mapping " is surjective, Dedekind appears to assume that M is

a subfield of Q. However, if M contains transcendental elements (over Q), then the permutation may notbe onto.

35Hence A is the fixed field of A.

29

the identity permutation A0 of A. Moreover, it follows from theorem II (in § 2) that the

group A is also the collection of all multiples of A0 with respect to M .36

One can convince oneself easily that theorem VI along with theorem II (in § 2)

fully encompasses Galois theory. To go into this in more detail, we assume that the

above group E is finite,37 from which it of course follows that all the groups A contained

in E are also finite. Let E be the field of the group E, and E0 its identity permutation,

then by VI M is a finite multiple of E, the residue of E is E0, and conversely,38 E is the

set of all multiples of E0 with respect to M . The key result of Galois theory is that on one

hand, the fields A which are divisors of M and at the same time multiples of E and, on

the other hand, the groups A contained in E, are in biunique correspondence.39 Firstly,

each such group A possesses a particular field A, which consists of all the numbers in

the field M which are single valued with respect to A,40 thus is a divisor of M , and since

each number in E, thus single valued with respect to E and also with respect to A, so A

is also a multiple of E. Furthermore, since by VI A is the set of all the multiples of the

identity permutation A0 of A with respect to M , it follows that two distinct groups A

possess two distinct fields A, too. Secondly, we have hence only to show that conversely

every field A which is a divisor of M and a multiple of E is also actually the field of a

group A contained in E. First, it follows from the statement (M,E) = (M,A)(A,E)

mentioned in § 1 that M is a finite multiple of A, whence by theorem II (in § 2) the

degree (M,A) is also the number of all those permutations ↵ of M which are multiples

of the identity permutation A0 of A, and moreover A is the field and A0 the residue

of the system A of these permutations ↵. Thus we need only to prove that this system

A is a group contained in E. Since A is a multiple of E, so that E0 is the divisor of36The proof given in (D.) of this theorem anticipates the modern proofs of Artin [1]. As Dean [2] notes,

for example, the independence of the automorphisms over M , whose proof can be found in (D. §161, lastparagraph), is used.

37More precisely, the assumption here is that E is a finite subgroup of the automorphism group of M .38Unclear whether umgekehrt is meant here in a vernacular or logical sense; we chose the logical sense.39“...sich gegenseitig eindeutig entsprechen.”40The field A is thus the fixed field of A.

30

A0 with respect to E, it follows that each permutation ↵ is also a multiple of E0 and

consequently contained in the group E. Furthermore, for every number x of the field A,

we have x = xA0 = x↵, thus also x↵�1= x, and if ↵1,↵2 are two such permutations

↵, it then also follows that x(↵1↵2) = (x↵1)↵2 = x↵2 = x, and therefore the resultant

↵1↵2 belongs to the same system A, which is consequently a group, q.e.d.

From the correspondence between the fields A and the groups A just proved, the

other theorems of Galois theory, which involve relations between several such fields A

and the corresponding groups A, follow immediately (D. § 166). We need not go into all

of this, as it is sufficiently well known, and we needed the above only to call attention

to the divergent behavior of infinite permutation groups.

§ 6.

Infinite Groups of Permutations.

We have seen that the field of all algebraic numbers H has infinitely many

permutations !, and that it [H] is mapped into itself by all of them; these permuta-

tions ! therefore form an infinite group, which we denote by G, and we ask whether

here as well there is a one-to-one correspondence between the algebraic fields A (the

divisors of H) and the groups A contained in G.

If we start with an arbitrary algebraic field A, and denote by A0 its identity

permutation, then by the theorem V (in § 4), which completely replaces theorem II (in

§ 2), there always exist permutations ↵ of the field H which are multiples of A0; be

the number (H,A) finite or infinite, it is always the case that A is the field and A0 the

residue of the system A of these permutations ↵. Furthermore, since A0 is the identity

permutation, it follows easily (as at the end of § 5) that this system A contained in G

is a group; we will call it the identity group of the field A.41 Furthermore, as already41The “identity group” of A is thus simply the subgroup of G leaving A fixed element-wise.

31

observed, since A is the field of A, it follows that two distinct fields A have distinct

identity groups A. The question raised above can therefore be answered affirmatively,

if one could prove that every group A contained in G is the identity group of a field A,

which is actually the case for finite groups A by the theorem VI (in § 5). Now to be

sure each infinite group A has a certain42 field A [namely, the field of A],43 which being

a divisor of H is thence algebraic, and since the identity permutation of H belongs to

A, it follows that the residue of A is surely the identity permutation A0 of this field A,

therefore A contains only those permutations ↵ of H which are also contained in the

identity group A0 of A. But the proof is lacking that conversely every permutation ↵0 in

A0, i.e., every multiple of A0 with respect to H , is also contained in the given group A;

or in other words, that the fields of two different groups are distinct. At first I regarded

this as most likely valid, and then after several futile attempts to prove it, I succeeded to

convince myself of the fallacy of this supposition by an example which I now want to

present to end this article.

This example is not related to the full field H , but rather to the simplest class

of infinite cyclotomic fields. If p is a fixed prime number, then each natural number n

corresponds to a root of unity

un = cos

2⇡

pn+ i sin

2⇡

pn. (1)

and we denote by Pn the cyclotomic field R(un) generated by this [root], which is of

degree '(pn) = (p� 1)pn�1, while P0 is the field R of rational numbers. From

un = upn+1, (2)

42 “...bestimmten...”43I.e., “The field of A.” We have A ✓ Aut(H/HA

) = A0

where H $ (1) and HA$ A.

32

it follows that in the infinite chain of fields

P0, P1, P2, P3, . . .

every term Pn is a divisor of the immediately succeeding Pn+1, and thence also of every

succeeding term Pn+s.

If any arbitrary chain of fields Pn is considered, which satisfy this last property,

then their least common multiple M can contain no other numbers t than those which

belong already to at least one field Pn and thus all succeeding fields Pn+s. For the set

of all these numbers t surely form, as is easily seen, a field which obviously is a divisor

of M , but is at the same time also a multiple of all Pn, therefore also a multiple of M ,

whence = M . One can therefore usefully denote this multiple by P1. If now " is any

permutation of M and "n the divisor of " with respect to Pn, then there exists an infinite

chain of permutations

"0, "1, "2, "3, . . . ,

in which each term "n is the divisor of all the succeeding terms "n+s. Conversely, if

a particular chain of permutations "n of the fields Pn is given, which satisfies this last

property, then it follows easily from the above mentioned constitution of the field M

(as in § 3), that there is a unique permutation " of M which is a multiple of all these

permutations "n.

If we apply this to our case of cyclotomic fields Pn = R(un), and denote by E

the set of all permutations " of their smallest multiple M = P1, then the divisor "n of "

with respect to Pn is completely determined by the conjugate number un"n = un" of un,

and, as is well known, this is always a power of un the exponent of which is a number

not divisible by p and which may be replaced by any number congruent to it modulo pn.

33

Let us denote this number by (n, "), and so it follows that

un"n = un" = u(n,")n , (3)

and since by (2) also

un" = (un+1")p,

hence

u(n,")n = u

p(n+1,")n+1 = u(n+1,")

n ;

therefore

(n, ") ⌘ (n+ 1, ") (mod pn), (4)

and this congruence expresses that "n is a divisor of "n+1. Conversely, if a chain of such

numbers (n, "), which are not divisible by p, is given, which satisfies the conditions (4),

then it follows from the general remarks above that it corresponds to a unique permuta-

tion " of M , which is determined by (3). With this one may decree that (n, ") should be

positive and less than pn, and if one sets (n + 1, ") = (n, ") + cnpn, then 0 cn < p ;

every arbitrarily chosen infinite sequence of such numbers c1, c2, c3, . . . together with

each of the p�1 numbers (1, ") produces a definite permutation ", and whence it follows

that the set E of all " form in a certain sense a continuous manifold, into which we shall

not go any further.44 45

As is well known, the field Pn is mapped into itself by every permutation, so

hence Pn" = Pn"n = Pn, and from this it also obviously follows that M" = M ;

therefore the set E forms a group (by § 5). If ", "0, thus also ""0, are contained in E, then44Dedekind seems to be anticipating the development of topological groups here. This is not a new

phenomenon for him. In his publication Stetigkeit und irrationale Zahlen, he observes that the operationson R “possess a certain continuity”, cf. [3], Vol. III, page 331, the last two paragraphs. This is apparentlythe first ideas of a topological group or ring and is the germ of algebraic topology, cf. Dugac’s article in[12], pp. 134-144.

45W. Krull [7] mentions that this remark is a fundamental idea which inspired him to endow a topologyon automorphism groups.

34

it follows from (3) that

un(""0) = (un")"

0= (un"

0)

(n,")= u(n,")(n,"0)

n ,

therefore [also]

(n, ""0) ⌘ (n, ")(n, "0) (mod pn) , (5)

and since the right side is not changed by switching ", "0, it follows that

""0 = "0", (6)

so therefore E is an abelian group.

Every permutation " generates by repeated composition with itself and its inverse

"�1 the sequence of all powers "r, which forms a group that we denote by ["].46 Of some

interest now is the question of whether there exist, aside from the identity permutation

M0 of M , which by itself forms a group, other finite permutations; i.e., [whether] there

exist such permutations ↵ which generate a finite group [↵].47 If m denotes the number

of distinct permutations ↵r in such a group [↵], then ↵m= M0 as is well known, and

conversely, if a natural number m satisfies this condition, then it follows from this that ↵

is finite. This requirement thus expresses by (3), (4), (5) that for every natural number

n, ↵ must satisfy the conditions

(n,↵)m ⌘ 1, (n,↵) ⌘ (n+ 1,↵) (mod pn). (7)46Of course this is simply the cyclic group generated by ".47Hence of finite order.

35

If we disregard the case p = 2, then it follows by close examination, which presents no

great difficulty, that there are only p� 1 finite permutations ↵; these are determined by

(n,↵) ⌘ (1,↵)pn�1

(mod pn), (8)

and one obtains all of these, when one lets (1,↵) run over an arbitrary complete system

of incongruent numbers modulo p, which are not divisible by p ; the corresponding

numbers (n,↵) form all p� 1 roots xn of the congruence

xp�1n ⌘ 1 (mod pn), (9)

and from this it follows that these p� 1 permutations ↵ satisfy the condition

↵p�1= M0. (10)

They form a group A and if one chooses for (1,↵) a primitive root modulo p, then

this group A = [↵]. Furthermore, if A denotes the field of A, then it follows from the

theorem VI (in § 5), that (M,A) = p� 1, therefore M is a finite multiple of A *).48

Now it is also not hard to find all finite and infinite divisors of the field M and to

determine the corresponding identity groups contained in E. For reasons of brevity, we

do not carry this out,49 and to conclude we wish merely to produce an example of the

proof promised above, that not every group in E is an identity group, or in other words,

that two distinct groups can have the same field.48*) In the case p = 2 omitted above, one easily finds that there are two finite permutations of M ,

namely the identity and one other, which is determined by mapping each root of unity un

to u�1n

.49Determination of the divisors of M was included as an appendix to Dedekind’s article in his collected

works and is likewise included at the end this translation.

36

We denote by g a fixed choice of primitive root modulo all powers of the (odd)

prime number p and define a permutation � of our field M by the congruence

(n, �) ⌘ g (mod pn),

for every natural number n, from which the existence condition (4) is satisfied. If again

�n denotes the corresponding divisor of � with respect to Pn, then

un�n = ugn, un�

rn = ugr

n

holds, and since the powers

g, g2, g3, . . . , g�(pn)

form a complete system of incongruent numbers modulo pn not divisible by p, the

powers

�n, �2n, �

3n, . . . , �

�(pn)n

exhaust all the permutations of the finite field Pn. Thus by a well-known theorem (or by

II in § 2) each number t contained in Pn, which satisfies the condition t� = t, hence also

the conditions t�r= t, must be rational, and so must belong to the field R. Now we

return to the permutation � of the field M , consider the group B = [�] consisting of all

powers of �, and look for its field, i.e., the set B of all numbers t of M single valued with

respect to B.50 This single-valuedness is completely determined by the requirement that

t� = t, because thus from this t��1= t and generally t�r

= t follow. Now since, as

noted earlier, every number t of the infinite field M surely also belongs to a finite Pn,

whence t� = t�n, so then must t also satisfy the condition t�n = t and consequently be

rational; therefore B = R. But on the other hand, it is clear that the identity group of R,

i.e., the set of all multiples of the identity permutation R0 of M , is the full group E of50Hence B is simply the fixed field of B.

37

all permutations " of M , and that R is the field of the group E, because every number

single valued with respect to E must be single valued with respect to B. Finally, that

the group B contained in E is distinct from E already follows from the fact that of the

p � 1 finite permutations ↵ determined above, only a single one, namely the identity

permutation M0 , is contained in B. Therefore, the two distinct groups B and E have

the same field R, q.e.d.

38

Supplement from Dedekind’s Nachlaß

Determination of the Divisors of M and their Identity Groups.

We now find all the divisors D of M . If D contains a number µ of exponent

ps (where s � 1) *),51 then because D is a multiple of R = As · Qe **),52 53 also a

multiple of As. Thus if there exist numbers µ in D, whose exponent ps exceeds every

given value, then D is a common multiple of all As and consequently also an extension

of A, whence

(M,A) = (M,D) · (D,A) = p� 1

(D,A) = e, (M,D) = f ; p� 1 = e · f

and consequently (easy)

D = A ·Qe .

In the contrary case, D is not a multiple of A; then there exists a number s such

that As is a divisor of D, but As+1 is not a divisor of D; whence the exponents of all the

numbers contained in D are ps, i.e.,D is a divisor of Ps = As ·P1, and therefore D is

a finite field,

D = As ·Qe. [ In the case that s = 0, D = R. ]

51*) Every number µ in M has a uniquely determined exponent ps, i.e., it is contained in Ps

, but not inPs�1.

52**) If e is any divisor of p� 1 = e · f , then denote by Qe

the field of degree e contained in P1; henceall the finite fields which are contained in M are of the form

As

·Qe

[s = 1, 2, . . . ],

where As

is the intersection of the fields A with Ps

.53The field A is the fixed field of the torsion subgroup A of E mentioned in § 6.

39

Identity Groups of the Subfields54 of M .

field Ms,e = As ·Qe ; p� 1 = e · f.

identity group Es,e : All permutations " of M , which are

multiples of the identity permutation of As ·Qe.

un" = u(n,")n ; (n, ") ⌘ (1, ")p

n�1(mod pn), 55

(s, ")f ⌘ 1 (mod ps).

field A ·Qe ; identity group E1,e :

un" = u(n,")n ; (n, ") ⌘ (1, ")p

n�1(mod pn),

(1, ")f ⌘ 1 (mod p).

54sic, as the original reads “Identitatsgruppen der Unterkorper von M”55This misprint appears in the Collected Works. It should read simply (n, ") ⌘ (n+ 1, ") mod pn.

40

German expression English translation Modern terminologyPermutation permutation (field) isomorphismR field of rational numbers QX field of real numbers RZ field of complex numbers CH field of algebraic numbers QDivisor divisor subfieldMultiplum multiple extension (field)reducibel reducible independentSystem system setInbegriff set set of elements such that . . .endlich Korper finite field finite degree extension(B,A) degree of B w.r.t.A [AB : A]ein Teil on a part of a subset ofA0 identity map on A idA or ◆AIdentitatsgruppe von A identity group of A associated group of A or

group belonging to field Aun primitive pn-th root of unity ⇣pnResultante resultant composition (of mappings)

Table 3.1. German expressions and terms used by Dedekind

41

REFERENCES

[1] Artin, E. Galois Theory, 2nd ed., North State, Hammond, 1964.

[2] Dean, E. Dedekind’s treatment of Galois theory in the Vorlesungen, TechnicalReport No. CMU-PHIL-184, Carnegie Mellon, Pittsburgh, 2009.

[3] Dedekind, R. Gesammelte mathematische Werke, I-III, Vieweg Verlag, Braun-schweig, 1930-1933. Reprinted by Chelsea Publishing Co., New Year, 1969.

[4] Dirichlet, L., Dedekind, R. Vorlesungen uber Zahlentheorie, 4th ed., ViewegVerlag, Braunschweig, 1894. Reprinted by Chelsea Publishing Co., New Year,1968.

[5] Hawkins, T. The Origins of the Theory of Group Characters, Arch. Hist. ExactSciences 7, No. 2 (1971), 142-170.

[6] Kiernan, B. M. The Development of Galois Theory from Lagrange to Artin,Arch. Hist. Exact Sciences 8, (1971/72), 40-154.

[7] Krull, W. Galoissche Theorie der unendlichen algebraischen Erweiterungen,Mat. Ann. 100, (1928), 687-698.

[8] Lang, S. Algebra, Revised 3rd ed., Springer, New York, 2002.

[9] Neukirch, J. Algebraic Number Theory, Springer Verlag, Berlin, 1999.

[10] Ostrowski, A. Uber einige Fragen der allgemeinen Korpertheorie, Journ. f.Math. 143, Heft 4, (1913), 255-284.

[11] Ramakrishnan, Dinakar, Valenza, Robert J. Fourier Analysis on Number Fields,Springer-Verlag, New York, 1999.

[12] Scharlau, W., Richard Dedekind 1831-1981, Vieweg Verlag, Braunschweig,1981.

[13] Steinitz, E. Algebraische Theorie der Korper, Journ. f. Math. 137, Heft 3,(1910), 167-309.

[14] Weber, H. Lehrbuch der Algebra, 2nd ed., vol. 1, (1898), Vieweg Verlag, Braun-schweig.

42

APPENDIX

RELATED PROOFS

A.1 Introduction

In this appendix we provide proofs of properties and theorems mentioned in

Chapter 2. As we aim to detail Dedekind’s construction of the correspondence of

the subfields of the Galois cyclotomic field ⌦ = [

1n=1Q(⇣pn) over Q and the closed

subgroups of its associated Galois group G = Gal(⌦/Q), we first characterize the

finite cases Gal(Q(⇣pn/Q) from which the infinite case is constructed (Section A.2).

From a modern perspective the relevant properties of the group G are best expressed

with respect to a topology on G, so we next (Section A.3) study the Krull topology,

which quite naturally yields (Section A.4) a revised form of the Fundamental Theo-

rem of Galois Theory (applicable to finite and infinite degree Galois extensions alike),

and with it, consideration of profinite groups. The relation of G as a profinite group

to the finite groups Gal(Q(⇣pn)/Q) is then clarified using the notion of a projective

limit (Section A.5). As G is topologically isomorphic to the multiplicative group Z⇥p

of p-adic units, we establish basic properties of the ring Zp and its profinite topology

(Section A.6). We then (Section A.7) characterize the closed subgroups of Z⇥p , which,

by the Fundamental Theorem of Galois Theory, correspond one-to-one to the subfields

of ⌦. Finally, we put the pieces together (Section A.8), linking up the modern analysis

with Dedekind’s notation, nomenclature and description of the correspondence, as given

in the Supplement (in [3]) to the article.

Note: Sections A.3-A.5 of this Appendix provide detailed readings of aspects of

Chapter 4 of Neukirch [9].

43

A.2 The structure of Gal(Q(⇣pn)/Q) for p odd and even.

We recall the structure of Gal(Q(⇣pn)/Q) for prime p, odd and even. Setting

⇣pn=e2⇡ipn (n2N), note that (Z/pnZ)⇥'Gal(Q(⇣pn)/Q), given by, e.g., a 7!�a, where

a = a+ pnZ and �a :⇣pn 7! ⇣apn . Clearly |(Z/pnZ)⇥|=�(pn).

Proposition A.1. For p an odd prime, (Z/pnZ)⇥ ' Cpn�1⇥Cp�1, where Ck denotes the

cyclic group of order k.

Proof. For p an odd prime, pn is known to have a primitive root a, so immediately

(Z/pnZ)⇥ =< a >' C�(pn). But as �(pn) = pn�1(p� 1) with p� 1 and pn�1 relatively

prime, we have by either Sylow or consequences of the Fundamental Theorem of Finite

Abelian Groups, C�(pn) ' Cpn�1⇥ Cp�1.1

Alternatively, we may determine an explicit isomorphism between (Z/pnZ)⇥

and Cpn�1⇥ Cp�1. First, taking primitive root a modulo pn, note that

a = apn�1

·

a

apn�1 = apn�1

· (apn�1�1

)

�1

where clearly apn�1 has order p� 1 and so is a generator for Cp�1. Also note that

(apn�1

)

pn�1= ap

n�1(p�1)(pn�1+···+1)= (a�(p

n))

(pn�1+···+1)= 1,

which must be of order pn�1, again as a is a primitive root modulo pn. Hence

(apn�1�1

)

�1 is a generator for Cpn�1 . Thus consider the natural map given by the

action a 7! (apn�1

, (apn�1�1

)

�1). That is a surjective homomorphism is immediate

from above. To show injective, note that a 2 ker implies apn�1

= apn�1�1

= 1.

But (pn�1, pn�1� 1) = 1 forces a = 1. Hence is an isomorphism such that, given

generator a of (Z/pnZ)⇥, we find Cpn�1=< ap

n�1�1 > and Cp�1 =< apn�1

>.1Or: Let C

p

n�1=< x >, C

p�1 =< y > for x, y 2 (Z/pnZ)⇥. Then (pn�1, p � 1) = 1 implies|xy| = lcm(pn�1, p�1) = �(pn). I.e., < xy >= C

�(pn). The result now follows by order considerationsand the definition of internal direct product.

44

We further note that a single generator may be used for Cpn�1 regardless of n,

namely p+ 1. For observing that (1 + p)pk= 1 + pk+1

+ pk+2↵, where ↵ denotes all

outstanding terms, we see (1+ p)pk⌘ 1+ pk+1

mod pk+2 for each k 2 N. In particular,

(1 + p)pn�1⌘ 1 + pn mod pn+1, which is ⌘ 1mod pn. Thus the order of 1 + p divides

pn�1. However, (1 + p)pn�2⌘ 1 + pn�1

mod pn and consequently (1 + p)pk6⌘ 1mod pn

for any degree k < n� 1. I.e., we have the

Proposition A.2. For odd prime p, ⇣pn=e2⇡ipn (n 2 N), and primitive root a modulo p,

Gal(Q(⇣pn)/Q) ' Cpn�1⇥ Cp�1 '< 1 + p > ⇥ < a >.

We immediately exhibit the Galois correspondence between subfields of Galois

extension Q(⇣pn)/Q and subgroups of Gal(Q(⇣pn)/Q), taking advantage of the expres-

sion of the Galois group as a direct product of cyclic groups. Note the lattice substructure

contributed by Cpn�1 is one-dimensional, being 1 Cp Cp2 · · ·Cpn�2 Cpn�1 .

However, the dimension of the substructure contributed by Cp�1 will be determined by

the number of distinct prime factors of p�1. For arbitrary factor (not necessarily prime)

e of p� 1, let He be the unique cyclic subgroup of Cp�1 of index e, so that He ' C p�1e

.

Similarly, for positive integer m where 0 m n � 1, let Gpm be the unique cyclic

subgroup of Cpn�1 of index pm, so that Gpm ' Cp(n�1)�m . The sublattice generated by

He and Gpm is given by Figure A.1 (following page), with parallel segments having the

same index.

Formally, the corresponding subfield structure, setting K = Q(⇣pn) (over Q) and

denoting by KS the fixed field of subgroup S of Gal(Q(⇣pn)/Q), is given by Figure A.2

(following page), with parallel segments having the same degree.

We recall that K<1>= K, KCpn�1

= Q(⇣p), KCp�1·Cpn�1= Q, and KHe

=

Q(

P

�2He�(⇣p)). As we will not be more deeply concerned with these results, we forgo

further analysis.

45

Cp�1 · Cpn�1

Cp�1 ·Gpm He · Cpn�1

Cp�1 He ·Gpm Cpn�1

He Gpm

< 1 >p�1e

e

p(n�m)�1

pm

Figure A.1. Sublattice generated by He and Gpm

KCp�1·Cpn�1

KCp�1·Gpm KHe·Cpn�1

KCp�1 KHe·Gpm KCpn�1

KHe KGpm

K<1>

p�1e

e

p(n�m)�1

pm

Figure A.2. Corresponding sublattice of K = Q(⇣pn) (over Q)

46

However, we wish to sketch the corresponding case for the ordinary prime

case excluded in the above discussion. Toward that end, as we have already seen that

Gal(Q(⇣2n/Q) ' (Z/2nZ)⇥ we note that (taking cases n = 1, 2 as obvious)

Proposition A.3. For n 2 N with n � 3, (Z/2nZ)⇥ =<�1>·<1 + 2

2>' C2 ⇥ C2n�2 .

Proof. Clearly we cannot proceed as in the odd prime case, as the order of (Z/2nZ)⇥

does not admit a relatively prime factorization by which we could apply the Sylow

theorems. For n = 3, (Z/2nZ)⇥ is isomorphic to the (non-cyclic) Klein four-group, as

no element has order greater than two.2 Now observe, for n � 3, that | < 5> | = 2

n�2,

since (1 + 2

2)

2k=

P

j

�

2k

j

�

2

2j⌘ (1 + 2

k+2) mod 2

k+3 for all k, by induction on k.

Further, �1 62< 5> else �1 = 5

⌫2

n for some nonnegative integers ⌫, n, implying

�1 ⌘ 5

⌫2

n⌘ 1mod 4. The claim follows.

Applying the proposition, we can immediately construct the subgroup lattice of

Gal(Q(⇣2n/Q)) (see Figure A.3).

The corresponding formal lattice of subfields of K = Q(⇣2n) (over Q) is given

in Figure A.4, or, more substantively, in Figure A.5, where H1 = Q(i(⇣8 + ⇣�18 )), H2 =

Q(⇣8+⇣�18 ), I = Q(⇣2n�3

+⇣�12n�3), J1 = Q(⇣2n�2

(⇣2n�1+⇣�1

2n�1)), J2 = Q(⇣2n�1+⇣�1

2n�1),

K1 = Q(⇣2n�1(⇣2n + ⇣�1

2n )) and K2 = Q(⇣2n + ⇣�12n ).

2This follows from the fact that 2n has no primitive root, which is what this proof amounts to.

47

< �1,�5 >

< �5 >< 5 > < �1, 52 >

< 5

2 >

< �1, 52n�4 >

< �52n�4 >< 5

2n�4 > < �1, 52n�3 >

< 5

2n�3 > < �52n�3 > < �1 >

< 1 >

Figure A.3. Subgroup lattice of Gal(Q(⇣2n/Q))

48

K<5> K<�5> K<�1,52>

K<�1,�5>

K<52>

K<52n�4> K<�52n�4

> K<�1,52n�3>

K<�1,52n�4>

K<�52n�3>K<52n�3

> K<�1>

K<1>

Figure A.4. Corresponding formal lattice of subfields of K = Q(⇣2n) (over Q)

49

Q(i) H1 H2

Q

Q(⇣8)

Q(⇣2n�2) J1 J2

I

K1Q(⇣2n�1) K2

Q(⇣2n)

Figure A.5. Lattice of subfields of K = Q(⇣2n) (over Q)

50

A.3 The Krull topology of a Galois extension

We define a topology (the Krull topology) on an arbitrary Galois extension with the aim

of establishing a generalization of the Fundamental Theorem of Finite Galois Theory.

Let k be a field and ⌦/k a Galois extension of k. That is, for every a 2 ⌦, a is alge-

braic over k, the minimum polynomial of a over k splits completely in ⌦, and ⌦ is a

separable extension. Set G = Gal(⌦/k), the Galois group of ⌦/k, and define the set

B = {�Gal(⌦/K) : � 2 G,K ✓ ⌦, K finite Galois over k}.

Proposition A.4. B is a basis for a unique topology on G.

Proof. Recall that a basis B for a unique topology on non-empty set S is a subset of

the power set of S satisfying the conditions: (1) S = [B2BB; and, (2) For arbitrary

B,B02 B and � 2 B \ B

0 , there exists B002 B such that � 2 B

00✓ B \ B

0 .

In this case, we find:

(1) Setting K = k implies G = Gal(⌦/k) 2 B.

(2) Take B = ⌧ Gal(⌦/K), B0= ⌧

0Gal(⌦/K

0) with ⌧, ⌧ 0

2 G and K/k, K 0/k

finite Galois, and let � 2 B \ B0 . Then, as cosets, note that

B \ B0

= �Gal(⌦/K) \ �Gal(⌦/K0)

= �(Gal(⌦/K) \Gal(⌦/K0))

= �(Gal(⌦/KK0))

2 B.

We denote the resulting topological space simply by G where no confusion is likely

to arise. Open subgroups of G have the form Gal(⌦/K) where K ✓ ⌦ and K has

51

finite degree over k (cf. Prop. A.9). We observe several useful properties of the Krull

topology.

Proposition A.5. The open subgroups of G are also closed in the Krull topology.3

Proof. Each open subgroup is the complement of the union of its open cosets.

Proposition A.6. Let S be an arbitrary subgroup of G. Let ⌦S denote the fixed field of

S. Then S = Gal(⌦/⌦S), where S denotes the topological closure of S in G.

Proof. As by construction S ✓ Gal(⌦/⌦S), certainly S ✓ Gal(⌦/⌦S

). We wish to

show that Gal(⌦/⌦S) = Gal(⌦/⌦S

). Toward that end, let us pick � in the complement

of Gal(⌦/⌦S) arbitrarily. There must be some x 2 ⌦S moved by �. As ⌦ is Galois,

it surely contains some finite Galois subextension K/k containing x. Then, for every

⌧ 2 Gal(⌦/K), we have ⌧(x) = x whereas �⌧(x) 6= x. Suppose now there were some

⇢ 2 Gal(⌦/⌦S) \ �Gal(⌦/K). Then, for every z 2 ⌦S

\ K, z = ⇢(z) = �⌧(z) =

�(z) for some ⌧ 2 Gal(⌦/K). In particular, this would hold for z = x, contradicting

our assumption. Hence the sets are disjoint. As � is arbitrary, every element of G �

Gal(⌦/⌦S) is contained in some (basic) open neighborhood disjoint from Gal(⌦/⌦S

),

and so Gal(⌦/⌦S) is closed. Thus S ✓ Gal(⌦/⌦S

).

For the reverse inclusion, recall that4 S = {⌧ 2 G|S \ U 6= ; for all open

U 3 ⌧} and let � 2 Gal(⌦/⌦S). Clearly it will suffice to show that S has nonempty

intersection with every basic open set containing �. If ⌧ 2 S \ �Gal(⌦/K) with K/k

arbitrary finite Galois, then by coset considerations �Gal(⌦/K) = ⌧ Gal(⌦/K).

K/k finite Galois ensures that the compositum ⌦

SK/⌦S is also finite Galois5.

Since obviously K ✓ ⌦

SK, it follows by inclusion reversal that Gal(⌦/⌦SK) ✓

3This proposition is true in any topological group.4Since � 62 S iff � is in the open complement of S, which implies the existence of some nonempty

open set U with U \ S = ;; and, conversely, for � 2 U , U open with nonempty intersection with S, thecomplement of U is closed and contains S, hence also S, and so � 62 S.

5Since ⌦SK contains K, the generators of K over k will suffice to prove ⌦S Galois, and clearly[⌦

SK : ⌦

S

] [K : k] <1.

52

Gal(⌦/K) or �Gal(⌦/⌦SK) ✓ �Gal(⌦/K), the latter being a basic open neigh-

borhood of � (in the relative topology induced on Gal(⌦/⌦S)). Hence it suffices

to show S \ �Gal(⌦/⌦SK) 6= ; for arbitrary � 2 Gal(⌦/⌦S) and arbitrary

k ✓ K ✓ ⌦ (K/k finite Galois).

So now let K/k finite Galois be given and consider the canonical restriction

homomorphism � : S ! Gal(⌦

SK/⌦S) mapping ⌧ 7! ⌧ |⌦SK . We claim � is surjective.

Clearly �(S) is a subgroup of Gal(⌦

SK/⌦S). Further, for any ⌧ 2 S and x 2 ⌦S ,

⌧ |⌦SK(x) = x by definition, so⌦�(S) ◆ ⌦S . The reverse inclusion is given by definition.

Hence⌦�(S) = ⌦S . By the Fundamental Theorem of Finite Galois Theory, we thus have

�(S) = Gal(⌦

SK/⌦S). I.e., � is surjective.

Hence given any � 2 Gal(⌦/⌦S), there is a ⌧ 2 S such that ⌧ |⌦SK =

�|⌦SK or ⌧��1|⌦SK = id|⌦SK 2 Gal(⌦/⌦SK), or ⌧ 2 �Gal(⌦/⌦SK). Thus

S \ �Gal(⌦/⌦SK) 6= ; for any � 2 Gal(⌦/⌦S) and any K/k finite Galois. From this

and the above considerations, Gal(⌦/⌦S) ✓ S follows.

We next observe that G, endowed with the topology T resulting from basis B, is

a topological group; i.e., the group and inversion operators are continuous (where, for

the group operator, the topology is extended naturally to the Cartesian product G⇥G).

Proposition A.7. G is a topological group.

Proof. It will suffice to prove the continuity of the group multiplication operator (�) on

basis B ⇥ B of G⇥G and of the group inversion operator ( ) on basis B.

(1) For : Though �1(�Gal(⌦/K)) = Gal(⌦/K)��1, we have

Gal(⌦/K)��1= ��1

Gal(⌦/K) 2 B since Gal(⌦/K) C G (as K/k Galois).

(2) For �: Take arbitrary (�, ⌧) 2 �

�1(⌘Gal(⌦/K)). Note for �� 2

�Gal(⌦/K), ⌧ 2 ⌧ Gal(⌦/K) we have simply �(��, ⌧ ) = ��⌧ = �⌧⌧�1�⌧ =

�⌧(⌧�1�⌧) 2 �⌧ Gal(⌦/K) = ⌘Gal(⌦/K), again since Gal(⌦/K) C G. Hence

53

�(�Gal(⌦/K) ⇥ ⌧ Gal(⌦/K)) ✓ ⌘Gal(⌦/K) for every pre-image (�, ⌧) and so

continuity follows.

Proposition A.8. G is a compact Hausdorff topological group.

Proof. We will proceed first by way of two lemmas. Toward that end, consider

the product topology on the Cartesian product of finite Galois extensions, G =

Q

K/k finite Gal(K/k), where each finite extension is endowed with the discrete topology.

Clearly every projection map ⇡K0 : G ! Gal(K0/k), i.e., (�K)K 7! �K0 , is contin-

uous for each K0 since ⇡�1K0({�K0}) = {�K0} ⇥

Q

K 6=K0Gal(K/k) is open in G by

construction. Further, the set

S = {{�K0}⇥

Y

K 6=K0

Gal(K/k) : K0/k finite Galois, �K0 2 Gal(K0/k)}

is a subbasis for G, i.e., a collection of subsets of G such that the union of its elements

yields G (obviously G = [S2SS) and the collection of finite intersections of its elements

forms a basis for G (obvious as well). Also note that subbasis elements are closed in

G (components not consisting of the entire space are nevertheless closed since each

component space is discrete).

Now to the first lemma:

Lemma A.1. G is a compact, Hausdorff topological group.

Proof. Compactness is clear by Tychonov. Hausdorff is trivial since two distinct

elements are contained in open sets differing on some component. That G is a

topological group is also trivial.

For the second lemma, define6

P = {(�K)K 2 G : 8L ◆ K, L,K finite Galois over k, �L|K = �K}.

6Note P is lim �

K

Gal(K/k), the projective limit of the finite Galois groups over k. Cf. Section 4.5.

54

P is made into a group in the natural way, and into a topological space by endowing it

with the relative topology of G (a subbasis consists of elements P \ S, P 2 P, S 2 S).

Lemma A.2. P is a compact, Hausdorff topological group.

Proof. The latter two properties are inherited from G. For compactness, it suffices to

show that P is closed in G. Toward that end, let L0◆ L be finite Galois extensions of

k and define the set ML0/L = {(�K)K 2 G : �L0|L = �L}. Immediately observe that

P = \L0/LML0/L, so it suffices to show ML0/L is closed.

By the isomorphism extension theorem, L0◆ L ensures that elements of

Gal(L0/k) are extensions of elements of Gal(L/k). We find (cf. Figure A.6)

that Gal(L0/k) = [

ni=1Si, where Si = [

mj=1{�i,j} such that �i,j|L = �i and

m = [Gal(L0/k) : Gal(L

0/L)].

k

L

L0

⌦

n

m

Figure A.6. Intermediate field extension tower in ⌦/k

Hence, setting K =

Q

K 62{L,L0}

finite Galoisover k

Gal(K/k), we find

ML0/L =

n[

i=1

({�j}⇥ Si ⇥K) =

n[

i=1

m[

j=1

({�j}⇥ {�ij}⇥K) ,

the finite union of an intersection of subbasis elements, each of which is closed. I.e.,

ML0/L is a finite union of closed sets in G, thus closed.

Finally, to conclude the proof of the proposition, consider the natural mapping

h : G ! P ✓ G given by � 7! (�|K)K . As h is a restriction mapping, it is clearly

55

a homomorphism. The kernel of h is trivial as � 2 kerh if and only if �|K = id for

every K, which is the case only if � = id 2 G. That h is onto is also immediate,

since for arbitrary (�K)K 2 P we may define � on ⌦ by �(a) = �K(a) for every

a 2 K. To show � is well-defined, let a 2 L where L/k is also finite Galois and note

�K(a) = �K |K\L(a) = �K\L(a) = �L|K\L(a) = �L(a). It is also easy to see that

� 2 G. Hence h is a group isomorphism.

Further, h is also readily seen to be a topological homeomorphism (i.e., a contin-

uous mapping with continuous inverse, also a continuous and open map): Let K0 be

some fixed finite Galois extension over k. Set K0=

Q

K 6=K0Gal(K/k) and notice that

h�1⇣⇣

{�K0}⇥K0⌘

\ P⌘

= �Gal(⌦/K0)

where � 2 G is chosen (by an application of Zorn’s lemma) such that �|K0 = �K0 .7

Hence h is continuous, and trivially also an open mapping (i.e., a map sending open sets

to open sets):

hh�1(({�K0}⇥K

0) \ P) = ({�K0}⇥K

0) \ P = h(�Gal(⌦/K0)).

A.4 The Fundamental Theorem of Galois Theory and Profinite Groups

We may now prove the Fundamental Theorem of Galois Theory:

Proposition A.9. Let⌦/k be a Galois extension of field k. Then the mapping given by

K 7! Gal(⌦/K) gives a bijection between intermediate fields k ✓ K ✓ ⌦ and closed

subgroups of G = Gal(⌦/k). Every open subgroup H of G has the form Gal(⌦/K) for

some finite degree extension K/k.7If ' 2 Gal(⌦/K0), then h(�') = (�'|

K

)

K

= �'|K0 ⇥ (�'|

K

)

K 6=K0 = �|K0 ⇥ (�'|

K

)

K 6=K0 .

56

Proof. First recall that G endowed with the Krull topology is a topological group.

To show Gal(⌦/K) is closed8 in G, assuming G 6= Gal(⌦/K), arbitrarily choose

� 62 Gal(⌦/K). Then there exists some element a 2 K such that �(a) 6= a. Take

the normal closure L� of k(a)/k, a finite Galois extension of k. Note that, for every

⌧ 2 �Gal(⌦/L�), we have, for some ' 2 Gal(⌦/L�), ⌧ = �' and so ⌧(a) =

�('(a)) = �(a) 6= a, whereas �(a) = a for every � 2 Gal(⌦/K) and a 2 K, and

hence �Gal(⌦/L�) \ Gal(⌦/K) = ;. As � 2 G � Gal(⌦/K) is arbitrary, it follows,

for some appropriate finite Galois extension L�/k in ⌦ chosen with respect to �, that

Gal(⌦/K)

\

2

4

[

�2G�Gal(⌦/K)

(�Gal(⌦/L�))

3

5

= ;

where[

�2G�Gal(⌦/K)

�Gal(⌦/L�)

is open, as a union of basic open sets. And clearly this latter union is precisely G �

Gal(⌦/K). Hence Gal(⌦/K) is closed in G.

That is injective is immediate: For let Gal(⌦/K) = Gal(⌦/L) and suppose

K 6= L. Without loss of generality, take a 2 L �K. Then there exists � 2 Gal(⌦/K)

such that �(a) 6= a, so � 62 Gal(⌦/L), contradiction.

That is surjective is not immediate: Let H < G and K = ⌦

H . Clearly

H ✓ Gal(⌦/K). Thence H ✓ Gal(⌦/K) = Gal(⌦/K) (cf. Proposition 4.6). For the

reverse inclusion, let L/k be finite Galois, � 2 Gal(⌦/K). Consider �Gal(⌦/L) \H .

We wish to show this intersection to be nonempty. Define a mapping � : Gal(⌦/K)!

Gal(KL/K) given by ⌘ 7! ⌘|KL. Figure A.7 suggests the motivation for consider-

ing �, recalling that as L/k is finite Galois, so is KL/K: By construction �(H) <

Gal(KL/K). Note, as K = ⌦

H , (KL)�(H)= K. For suppose a 2 KL�K: Then there

exists � 2 H such that �(a) 6= a, so �|KL(a) 6= a, implying �|K 62 Gal(KL/K). But as8This is obvious if K/k is finite by Prop. A.5.

57

[KL : K] < 1, we have, by the Fundamental Theorem of Finite Galois Theory, K =

(KL)Gal(KL/K). Now (KL)�(H)= (KL)Gal(KL/K), forcing �(H) = Gal(KL/K).

So � 2 Gal(⌦/K) implies �(�) = �|KL and, as �|KL 2 �(H), there exists ⌧ 2 H

such that ⌧ |KL = �|KL. This is the same as saying there exists ⌧ 2 H such that

⌧ |KL = �|KL, hence that ⌧ 2 H \ �Gal(⌦/KL) since, for every ' 2 Gal(⌦/KL), we

have �'|KL = �|KL(1G). But H \ �Gal(⌦/KL) ✓ H \ �Gal(⌦/L) since KL ◆ L.

Hence for every � 2 Gal(⌦/K) we find �Gal(⌦/L) \H 6= ;.

k

L

K

KL

⌦

Figure A.7. Composite lattice over a Galois & an arbitrary extension of k

To show H < G open, we can proceed in at least two ways:

(1) Note that G = [

�2G/H�H implies H also closed. Thus by above H =

Gal(⌦/K) for some K. Now as G is compact, the coset cover has a finite subcover.

Hence [G : H] is finite and immediately so is [K : k].

(2) If K/k is finite (not necessarily normal), take L to be its normal closure in

⌦. As L/k is finite Galois, Gal(⌦/L) is a basic open set. Now merely observe that

Gal(⌦/K) = [

�2Gal(⌦/K)�Gal(⌦/L), a union of basic open sets.

Thus the proposition follows.

We define (with Neukirch [9]) a profinite group as a Hausdorff compact topolog-

ical group with a basis of neighborhoods of the identity consisting of normal subgroups.

Proposition A.10. For profinite group G the set N = {N : N C G, N open} is a basis

of neighborhoods of 1G.

58

Proof. The proposition is nearly self-evident, as

(1) 1G 2 N (8N 2 N );

(2) 8M,N 2 N , N \M 2 N ; and

(3) For every open subset U of G containing the identity, there exists an N 2 N

such that N ✓ U .

We note here a result perhaps useful in the following:

Proposition A.11. A compact Hausdorff topological group is totally disconnected if and

only if it admits a basis of neighborhoods of the identity consisting entirely of normal

subgroups (i.e., if and only if it is profinite).

Proof. First recall that a topological space X is called disconnected if there exist

nonempty open sets U and V whose intersection is null and whose union is X , a space

that is not disconnected is called connected, a maximal connected subset of a space

is called a component, and a totally disconnected space is one in which only points

(single-element subsets) are connected.

In the following, let G be a compact Hausdorff topological group.

For sufficiency, let N = {Ni C G : i 2 I} for I an index set denote a basis

of neighborhoods of the identity 1G of G and let C(1G) denote the component of G

containing 1G. Suppose there exists � 2 C(1G)� {1G}. As the space is Hausdorff, � 62

Ni for some Ni 2 N . Since Ni is open implies Ni closed, both Ni and its complement

G � Ni are open in G. Hence we have C(1G) = (C(1G) \ Ni) [ (C(1G) \ (G �

Ni)), a disjoint union of nonempty open sets. But this implies C(1G) is disconnected;

a contradiction. Therefore C(1G) = {1G}. Since multiplication by a fixed element of

the group is a homeomorphism on G and �(C(1G)) = {�} for every � 2 G, sufficiency

follows.

For necessity, suppose G is totally disconnected. To show that G is a profinite

group, let N be the set of open normal subgroups of G. To show that N is a basis of

59

neighborhoods of 1G, we note first that 1G 2 N for every N 2 N . Secondly, certainly

N1, N2 2 N implies N1 \N2 C G.

Lastly, we need to show that for every open set U in G such that 1G 2 U ,

there exists some N 2 N such that N ✓ U . To show this, following [11], we first

show (Lemma A.3) that every open set U containing the identity contains some open

compact subset K containing the identity. This result relies on a technical result about

compact Hausdorff spaces that we relegate to a footnote for the sake of continuity of

exposition. We then proceed by two additional lemmas (A.4, A.5) to show that any such

open compact subset K must itself contain some element N of N .

Lemma A.3. Let U be an arbitrary open neighborhood of the identity in G (as given

above). There exists a compact open subset K of G containing the identity and

contained in U .

Proof. Let K1 denote the collection of compact open subsets of G containing the identity

1G. Certainly the set is nonempty as G itself is a compact open subset containing 1G.

Note that the set \K2K1K is connected in G.9 But as G is a totally disconnected space9Again, following [11], we show a more general result: Let X be a compact Hausdorff space and

let x 2 X . Define collection K

x

= {K : Kis a compact open subset ofX containingx}. The setY = \

K2Kx

K is connected in X . We first note that Kx

is nonempty as X is an element of the set. Theproof proceeds by contradiction: Suppose Y is a disconnected set in X . Then, for some disjoint relativelyclosed sets Y1 and Y2 in Y we have Y = Y1[Y2. But then, as Y is itself a closed set (as an intersection ofclosed sets), Y is compact, and hence each of Y1 and Y2 is compact in Y . However, a relatively compactsubset in a subspace of a compact Hausdorff space S is itself a compact subset of S. Hence Y1 and Y2 arecompact in X and thus closed. So we take Y = Y1[Y2 for disjoint closed sets Y1 and Y2 in X . Next, as acompact Hausdorff space is normal, there exist disjoint open sets U1 containing Y1 and U2 containing Y2.So now Y ✓ U1 [U2 and X � (U1 [U2) ✓ X � Y = Y c. Set Z = X � (U1 [U2). Trivially the set K

x

covers Y , so the set Kc

x

of complements in X of K 2 K

x

surely covers Y c and so also Z, which is a closedsubset of X , hence compact (as X is compact Hausdorff). Now Y c

= (\

K

K)

c

= [

K

Kc

◆ Z, so thecollection K

c

x

is an open cover of Z (since each K, as a compact set in a Hausdorff space, must be closed).Hence Kc

x

contains a finite subcover of Z, say {Kc

i

}

r

i=1. Then Z \ (\ri=1Ki

) = ;. Set W = \

r

i=1Ki

. Wis an open compact neighborhood of x. So W 2 K

x

. As Z \W = [X � (U1 [ U2)] \W = ;, we haveW ✓ U1 [ U2. So we may write W = (W \ U1) [ (W \ U2), where the union is disjoint. Now x 2 Wimplies either x 2 (W \U1) or x 2 (W \U2), say the former without loss of generality. Clearly W \U

i

(i = 1, 2) is open in X . Each is also compact, as any open cover C of e.g. W \ U1 can be extended to anopen cover C

0of W by taking C

0= C [ {{W \ U2}}, which admits a finite subcover, some elements of

which must be distinct from W \U2 and thus form a finite subcover of C. Hence W \U1 is in K

x

and soY ✓ W \ U1. Now W \ U1 ✓ Y1 implies Y ✓ Y1. But Y = Y1 [ Y2 ✓ U1 [ U2 (each union disjoint)and Y \ U2 = ; imply Y2 = ;, contradicting our supposition. Therefore, Y is connected in X .

60

and 1G 2 \K2K1K, we must have \K2K1K = {1G}. Now let U be an arbitrary open

neighborhood of the identity in G. Then G � U is a closed, hence compact, subset of

G. The set Kc1 of complements in G of K 2 K1 covers G � {1G}, and since G � U ✓

G�{1G}, the collection K

c1 covers G�U . But since every compact subset of a Hausdorff

space is closed, every element of K1 is closed. Hence K

c1 is an open cover of compact

set G� U and so admits of a finite subcover, say {Kci }

ri=1.

Next, as G�U ✓ [iKci , we find \iKi ✓ U . Set K = \iKi. As each Ki is open

and contains 1G, the finite intersection K is open and contains 1G. Also, as each Ki is a

closed set (hence compact), K is closed, hence compact. Thus K is a compact open set

containing 1G and contained in U .

Lemma A.4. Let K be a compact open subset containing 1G of the topological group G.

There exists a subset V containing 1G that is open, symmetric, and such that KV ✓ K.

Proof. Consider the map ' : K ⇥ K ! G given by (⌘,�) 7! ⌘�. For open U in G,

preimage '�1(U) = µ�1

(U) \ (K ⇥ K) where µ is the (continuous) group operation

for G. As K is open in G, K⇥K is open in G⇥G and so ' is continuous. In particular,

'�1(K) is open in K ⇥K. As (, 1G) 2 '�1

(K) for every 2 K, there exist sets U,

˜V open in G, contained in K, such that 2 U, 1G 2 ˜V. Let V =

˜V \ ˜V�1✓ K.

Clearly 1G 2 V. Further, (, 1G) 2 U ⇥ V ✓ '�1(K) implies 2 UV ✓ K. As

K = [

2KU, compactness implies K = [

ni=1Ui for some n 2 N. Set V = \

ni=1Vi ,

where Vi corresponds as given above to the Ui . Then note that V 3 1G, is open and

symmetric. Finally, since UiV ✓ UiVi ✓ K, we have KV = [

ni=1UiV ✓ K.

Lemma A.5. Let K be a compact open subset containing 1G of topological group G.

Then there exists N0 2 N such that N0 ✓ K.

Proof. Take H = [

1n=1V

n, where V is as given in Lemma A.4. That H is an open

subgroup of G is proved in the routine way: As V is open, so is vV for every v 2 V

(more generally, gV is homeomorphic to V for every g 2 G). Hence V 2, a union

61

of open sets, is open and, by induction, so is V n. Immediately it follows H is open.

Now V n= V �n since V = V �1 and so H = H�1 (H contains all inverses). Also,

for any x, y 2 H , there must be V m3 x and V n

3 y and so xy 2 V mV n✓ H

(closure and identity). I.e., H < G. Now, by the previous lemma, H ✓KH✓K. By

compactness, [G :H] = k <1 and so G=[

ki=1�iH for some �i 2G. To conclude, set

N0 = \�2G�H��1= \

ki=1�iH�

�1i . Hence N0 2 N and N0 ✓ K.

To conclude the proof of Proposition A.11: By Lemma A.3 open neighborhood

U of the identity of G contains some compact open subset K that contains the identity.

By Lemma A.5 every such compact open neighborhood K of the identity contains some

element N 2 N . Hence we have shown that for every open set U in G such that 1G 2 U ,

there exists some N 2 N such that N ✓ U .

A.5 Projective Limits, Relation to Profinite Groups

Let (I,) be a directed set, {(Xi, fij) : i, j 2 I, i j} a projective system over I.

For our purposes, we take the Xi to be topological spaces, the fij to be continuous

homomorphisms from Xi to Xj satisfying appropriate compatibility conditions; namely

fii = idXi , fij � fjk = fik for i j k. Define the projective limit of the Xi to be

the set X = {(xj)j2I : 8i j fij(xj) = xi}, where the Cartesian productQ

i Xi is

endowed with the product topology and X with the relative topology.10 We also denote

the projective limit as lim �

i

Xi.

Proposition A.12. If Xi is Hausdorff for every i 2 I, then (1) BothQ

i Xi and X are

Hausdorff, and (2) X is closed inQ

i Xi.

Proof. (1) is immediate. To show (2), first define Xij = {(xl)l 2Q

i Xi : fij(xj) =

xi whenever i j} and observe that X = \

ijXij . Hence it suffices to show Xij closed

10Recall the product space has a subbasis consisting of sets of the form Uj

⇥

Q

i 6=j

Xi

where Uj

isopen in X

j

.

62

for i j. Next, note that Xij = {(xl)l 2Q

i Xi : ⇡i((xl)l) = fij � ⇡j((xl)l)}, where

again all fij and projection mappings ⇡i, ⇡j are continuous.

The argument proceeds from the following generic topological property:

Lemma A.6. If X, Y are topological spaces, Y is Hausdorff, and f , g are continuous

maps from X to Y , then the set E = {x 2 X : f(x) = g(x)} is closed.

Proof. We show that the complement X � E is open. Choose any z 2 X � E. Then

f(z) 6= g(z) and there exist disjoint open sets U 3 f(z) and V 3 g(z), as X is Haus-

dorff. Now z 2 f�1(U) \ g�1

(V ) = Kz, which is open in X (continuity of f and g).

But f(Kz) ✓ U and g(Kz) ✓ V imply Kz \ E = ;. As z 2 X � E is arbitrary, E is

open and the lemma follows.

The claim now follows immediately.

Proposition A.13. If Xi is a nonempty compact Hausdorff space for every i 2 I, then

the projective limit X is compact and nonempty.

Proof. As the product spaceQ

i Xi is compact (by Tychonov) and X is closed inQ

i Xi,

X is compact. Now suppose X = ;. By the finite intersection property on compact X ,

there exist i1 j1, · · · , in jn such that � = \

nk=1Xikjk = ;. But observe: Choose

k 2 I such that j⌫ k for all ⌫ = 1, . . . , n. As Xk 6= ;, we can choose some xk 2 Xk.

Choose element (xl)l 2Q

i Xi where xl = flk(xk) if l k and xl is chosen at will when

l 6 k. Note that i⌫ j⌫ k and fi1j1(xj1) = xi1 . By compatibility,

fi1j1(xj1) = fi1j1(fj1k(xk)) = fi1k(xk) = xi1 .

Thus (xl)l 2 �, contradicting the conclusion drawn from the supposition using the finite

intersection property.

Proposition A.14. (“Universal Property”): Let {Gi, gij}ij be a projective system of

topological groups, with each gij a continuous group homomorphism. Let G = lim

�

i

Gi

63

be the projective limit of the Gi. For each i, define gi = ⇡i|G. Suppose H is a topological

group and, for each i, suppose hi : H ! Gi is a continuous homomorphism such that

the diagram

H Gj

Gi

hi gij

hj

commutes (hi = gij � hj). Then there exists a unique continuous homomorphism h :

H ! G such that the diagram

H G

Gi

hi gi

h

commutes (hi = gi � h) for each i.

Proof. For � 2 H , define h :H!G by � 7! (hi(�))i2I 2Q

i Gi. If i j, then note

hi(↵) = gij(hj(↵)), so h is a group homomorphism. Also, gi � h(�) = gi((hi(�))i2I) =

hi(�) for all i 2 I.

Uniqueness follows trivially from the assumed commutativity of the first of the

two above diagrams: If h and f both satisfy the requisite conditions, then hi = gi � f =

gi � h for every i 2 I.

For the continuity of h : H ! G, take subbasis element A = G\ (Ui0 ⇥Q

i 6=i0

Gi)

of G, with Ui0 ✓ Gi0 open. Then note: � 2 h�1(A) holds if and only if [ gi0h(�) 2 Ui0

holds, which holds if and only if] hi0(�) 2 Ui0holds, which holds if and only if � 2

h�1i0(Ui0) holds. Thence h�1

(A) = h�1i0(Ui0). Continuity follows.

Proposition A.15. Given G a profinite group, N = {N < G : N C G open}. Then

G ' lim

�

N2NG/N , where the isomorphism is both algebraic and topological.

64

Proof. As (N,◆) forms a directed set with M N iff M ◆ N , consider map

f : G! lim

�

N2NG/N = {(�NN)N : 8M N �MM = �NM}

given naturally by � 7! (�N)N2N. Note as G/N ! G/M by �N 7! �M for all

M N , the map f is well-defined. Clearly f is a group homomorphism. Further,

as the quotient topology on G/N is here equivalent to the discrete topology, we may

endow lim

�

N2NG/N with the relative topology of G. The map is one-to-one since � 2 ker f

implies � 2 \N2N

N = {1G} (as G is Hausdorff).11

To show f continuous, take profinite-limit basis element (writingQ

N

0 forQ

N 62{N1,...,Nk})

U =

{⌧1N1}⇥ {⌧2N2}⇥ · · ·⇥ {⌧kNk}⇥

Y

N

0G/N

!

\ lim

�

N2NG/N

and consider f�1(U) in G: If U = ;, certainly f�1

(U) is open in G; if U 6= ;, there

exists z = (�NN)N2N 2 lim

�

N2NG/N such that z = (. . . , �N0N1, . . . �N0Nk, . . .) for every

N0 such that Ni N0 for i = 1, . . . , k. (For instance, we may take N0 ✓ \ki=1Ni 2 N.)

Whence, as z 2 U , an open basis element in the profinite limit of the G/N , we have

�N0Ni 2 {⌧iNi} and thus ⌧iNi = �N0Ni for each i = 1, . . . , k. I.e., ⌧i = �N0 for

i = 1, . . . , k. Hence

U =

{�N0N1}⇥ {�N0N2}⇥ · · ·⇥ {�N0Nk}⇥

Y

N

0G/N

!

\ lim

�

N2NG/N.

Now certainly f�1(U) ◆ �N0

˜N where ˜N = \

ki=1Ni 2 N is open in G. For the reverse

inclusion, let � 2 f�1(U). I.e., f(�) = (�N)N2N 2 U . But (�N)N2N 2 U =

11As G is Hausdorff, if z 6= 1

G

were in \N then there would exist disjoint open neighborhoodsU1 3 1

G

and Uz

3 z. There then would exist a fundamental neighborhood N of 1G

with 1 2 N ✓ U1.But then z 62 N would imply z 62 \N , contradicting supposition. The result follows.

65

(. . . , �N1, . . . , �Nk, . . .) implies �Ni = �N0Ni, which implies � 2 �N0Ni for all Ni,

which implies � 2 \ki=1�N0Ni = �N0\ki=1Ni = �N0

˜N . Hence we find f�1(U) ✓ �N0

˜N .

From the fact that f(G) is the continuous image of a compact set it follows that

f is a homeomorphism, so an open (closed) set is mapped to an open (closed) set. Hence

f(G) is closed. Furthermore it is Hausdorf, as f(G) ✓ lim

�

N2NG/N (the profinite limit is

Hausdorff).

Now to show f onto it would suffice to show f(G) dense in G. I.e., for every

z 2 lim

�

N2NG/N and every open set U 2 lim

�

N2NG/N containing z, U \ f(G) 6= ;.

For each such U there is some basic open set VU such that z 2 VU ✓ U . Recall

VU has the form ({⌧1N1} ⇥ · · · ⇥ {⌧kNk} ⇥Q

N

0G/N) \ lim

�

N2NG/N . It would suffice

to show VU \ f(G) 6= ; for every open set U . Now z = (�NN)N2N 2 VU implies

V = ({�1N1} ⇥ · · · ⇥ {�kN |k} ⇥Q

N

0G/N) \ lim

�

N2NG/N since ⌧iNi = �iNi (coset

equality). Choose N0 2 N such that Ni N0 (i.e., Ni ◆ N0 ) for i = 1, 2, . . . k. Then

�N0Ni = �NiNi and so V = ({�N0N1}⇥ · · ·⇥{�N0Nk}⇥Q

N

0G/N)\ lim

�

N2NG/N . Hence

f(�N0) 2 VU . I.e., f(�N0) ✓ V \ f(G) ✓ U \ f(G).

A.6 The p-adic integers Zp

The case Dedekind considers, namely the union of cyclotomic p-power extensions (for

fixed odd prime p), has Galois profinite group isomorphic to the unit group of the p-adic

integers. Hence, to describe Dedekind’s work in modern terms, we recall the structure

of profinite ring Zp and its unit group.

With directed set (N,) and canonical homomorphisms Z/pnZ ! Z/pmZ

(m n) as a projective system and topologizing in the usual way, we form the projective

limit

Zp := lim

�

n2NZ/pnZ = {(an + pnZ)n2N : 8m,n m n am ⌘ an mod pm}.

66

Proposition A.16. Zp is an additive abelian topological group.

Further,

Proposition A.17. By the natural mapping Z ,! Zp given by a 7! (a+ pnZ)n2N, Z has

dense image in Zp.

Proof. Consider basic open set U=({a1+pn1Z}⇥ · · ·⇥{ak+pnkZ}⇥Q

n

0Z/pnZ)\Zp

and assume it to be nonempty (Q

n

0:=

Q

N�{n1,n2,...,nk}). Without loss of generality, choose

a 2 Z such that12U = ({a+pn1Z}⇥ · · ·⇥{a+pnkZ}⇥

Q

n

0Z/pnZ)\Zp. Immediately

note that Z is dense in Zp.

Proposition A.18. F = {pnZp : n 2 N} is a fundamental system of neighborhoods of

the additive identity.

Proof. (1) That 0 2 N 2 F for every such N is clear; (2) Immediately we find pnZp \

pmZp = pmax(n,m)Zp; (3) For any open set U in Zp containing 0 there exists some basic

neighborhood of 0 of the form V = ({a+ pn1Z}⇥ · · ·⇥ {a+ pnkZ}⇥Q

n

0Z/pnZ)\Zp

for some a 2 Z, and thus

V=({pn1Z}⇥ · · ·⇥ {pnkZ}⇥Y

n

0Z/pnZ) \ Zp ◆ pmax{ni : i=1,...,k}Zp 2 F.

Next observe that Zp satisfies the nice property

Proposition A.19. Zp/pnZp ' Z/pnZ.

Proof. We prove the isomorphism by establishing a short exact sequence of ring

(module) homomorphisms 0 ! A ! B

'! C ! 0 where is injective, ' is

surjective, and im( ) = ker', which implies B/im( ) ' C. Toward that end,

consider the homomorphism sequence 0 ! pnZ i,! Z '

! Zp/pnZp ! 0 where

12 We may do such, e.g., simply by taking n � ni

for all i = 1, . . . , k, for then we have an

⌘ ai

mod pni (i = 1, . . . , k) by definition.

67

i is the natural injection mapping and a'7! (a + pmZ)m2N + pnZp. Note that

' � i acts as ai7! a = pnx

'7! 0, so im i ✓ ker'. For the reverse inclusion, if

'(a) = (a+ pmZ)m2N + pnZp 2 ker', i.e., if (a+ pmZ)m2N 2 pnZp, then we have, on

every component (i.e., for every m 2 N), a+ pmcm = pnam + pmdm for some integers

am, cm, dm. That is, a = pnam+pm(dm�cm), which is⌘ 0 mod pm for every m � n.

I.e., a 2 pnZ, so ker' ✓ im i and equality follows. Finally, to assure that ' is onto,

take arbitrary element z + pnZp 2 Zp/pnZp, recall that Z is dense in Zp, and note that

this by definition implies a 2 z + pnZp for some integer a.

Proposition A.20. Zp is a topological ring (Zp,+, ·).

Proof. Zp is made into a ring in the natural way. That it is a topological ring, there

is but to show that multiplication is continuous. Defining the map Zp ⇥ Zpµ! Zp by

(z, w)µ7! zw, immediately note that (z + pnZp, w + pnZp) 7! zw + pnZp.

Finally, we observe that

Proposition A.21. (Zp)⇥= lim

�

n2N(Z/pnZ)⇥.

Proof. For z 2 Z⇥p , there exists z�1

2 Z⇥p such that 1 = zz�1

= z�1z where z =

(am + pmZ)m2N for am 2 Z, z�1= (bm + pmZ)m2N for bm 2 Z. Now zz�1

=

(ambm+pmZ)m2N = (1+pmZ)m2N if and only if ambm ⌘ 1 mod pm for every m 2 N.

Hence, in particular am 2 (Z/pmZ)⇥ for every m 2 N, and thus, by compatibility,

z 2 lim

�

n2N(Z/pnZ)⇥. Conversely, if z 2 lim

�

n2N(Z/pnZ)⇥, then each component has an

inverse, and, by compatibility, this inverse lies in Z⇥p .

A.7 Closed Subgroups of Z⇥p

By the Fundamental Theorem of Galois Theory, closed subgroups of Z⇥p will corre-

spond bijectively to subfields of Gal(Q(⇣p1)/Q), both of which we wish to characterize,

following Dedekind. We now investigate closed subgroups of Z⇥p , for p odd. First, note:

68

Lemma A.7. For ↵ = (an + pnZ)n 2 Z⇥p , we have ↵ 2 1 + psZp () as ⌘ 1

mod ps.

Proof. ↵ = (an + pnZ)n 2 Z⇥p is such that, for every m n, am ⌘ an mod pm and

(a1, p) = 1. Hence ↵ 2 1 + psZp if and only if there exist c0, c1, . . . , cn, . . . 2 Z such

that

an =

8

>

>

<

>

>

:

1 mod pn : n s (compatibility down)

1 + ps(Pn�s�1

i=0 cipi) mod pn : n > s (compatibility up)

Hence ↵ 21 + psZp if and only if as ⌘ 1 mod ps.13

Proposition A.22. Let p be an odd prime. We have the topological isomorphism Z⇥p '

µp�1 · (1 + pZp),14 where µp�1 = {! 2 Z⇥p : !p�1

= 1}.

Proof. Define Z⇥p

'! µp�1 ⇥ (1 + pZp) by (an + (pn))n = ↵

'7! (!(a1),

↵!(a1)

),15 where

!(a1) = (apn�1

1 + pnZ)n. Note !(a1) 2 µp�1 for a1 6⌘ 0 mod p.

That ' is well-defined: For some b1 2 N, ↵!(a1)

= (b1 + (p), . . .) with b1 ⌘ 1(p).

As ↵ = (a1+(p), . . .), is a unit, immediately we have !(a1)�1= !(a01) for some natural

number a01 such that a1a01⌘1(p).

That ' is a homomorphism: Immediately by construction, '(↵�) =

(!(a1b1),↵�

!(a1b1)) = (!(a1)!(b1),

↵�!(a1)!(b1)

) = (!(a1),↵

!(a1))(!(b1),

�!(b1)

) = '(↵)'(�).

13As an aside, consider

a+ psZp

= {1 + ps� : � 3 Zp

} = {(1 + psbn

+ pnZ)n

: bm

⌘ bn

(pm) 8m n},

where � = (bn

+ pnZ)n

. Note that ↵ 2 (an

+ pnZ)n

2 1 + psZp

if and only if an

⌘ 1 + psbn

(pn),am

⌘ 1 + psbm

(pm) and bm

⌘ bn

(pm) (for every m n), if and only if an

⌘ 1 mod ps andam

⌘ an

(pm) (for every m n). For an

⌘ 1(ps) let an

= 1 + psbn

for some bn

(for each n)and a

m

= 1 + psbm

(for m n). Then as an

⌘ am

(pm+s

) we have 1 + psbn

⌘ 1 + psbm

(pm+s

) orbn

⌘ bm

(pm). Note now that by compatibility, and as as

⌘ 1 mod ps, we have ↵ = 1+p� or ↵�1p

= �.Explicitly, a

n

= 1 for n s, as+1 = b

s+1 arbitrary, and as+m

=

P

m

n+1 bs+n

pn with as+m

⌘ ak

(pk)for k < s+ n.

14In the excluded case where p = 2, we find Z⇥2 = {±1} · (1 + 4Z2).

15Note: < ↵ >=

↵

!(a1)is a principal unit in Z⇥

p

, i.e. 2 1 + pZp

.

69

That ' is injective: ↵ 2 ker' by definition if and only if both !(a1) = 1 and

↵!(a1)

= 1.

That ' is surjective: For arbitrary element (!(a1), 1 + p�) 2 µp�1 ⇥ (1 + pZp)

note that ↵ = !(a1)(1 + p�) maps to it.

That ' is a homeomorphism: For continuity, consider open set B =

{!(a1}⇥ (

↵!(a1)

+pNZp)} for some N 2 N. Since ↵+pN� 7! (!(a1),↵

!(a1)+pN �

!(a1)),

we have '�1(B) = ↵ + pNZp, an open preimage in Z⇥

p . That ' is an open mapping,

observe, for basic open set 1 + pNZp that '(1 + pNZp) = {1}⇥ (1 + pNZp).

Hence the claim is proven.

Corollary A.1. µp�1 is the torsion subgroup16 of Z⇥p .

Proof. That Tor(Z⇥p ) ◆ µp�1 is apparent. For the reverse inclusion, consider ↵ 2

Tor(Z⇥p ). There exist !, � such that ↵ = !(1+p�). Suppose ↵ 62 µp�1. Then 1+p� 6= 1;

i.e., � 6= 0. That is, � 2 pNZp � pN+1Zp for some N 2 N; i.e., � = pN� for some

� 2 Z⇥p . Since already ! 2 Tor(Z⇥

p ), we must have 1 + p� 2 Tor(Z⇥p ). Hence

(1 + p�)M = 1 for some M 2 N.

Now write M = mps for m 2 N, where (m, p) = 1 and s � 0 and note

(1 + p�)M = (1 + pN+1�)psm

= (1 + pN+s+1�0)

m for some � 02 Z⇥

p , by binomial

expansion. Again, by binomial expansion, (1 + pN+s+1)

m= 1 +mpN+s+1�

00 for some

�002 Z⇥

p . Hence 1 +mpN+s+1�00= 1, implying mpN+s+1�

00= 0, a contradiction.

Proposition A.23. Let t 2 N and � 2 Z⇥p . Then < 1 + pt� > = 1 + ptZp.

Proof. For ✓: We find < 1 + pt� >✓ 1 + ptZp by exhibiting an inverse of 1 + pt�.

The right-hand side is an open subgroup, thus closed, and so contains the closure of

< 1 + pt� >.

For◆: It suffices to show < 1+ pt� > is dense in 1+ ptZ. Let U be an arbitrary

open subset of 1 + ptZp and let 1 + pt� be an element of < 1 + pt� >. Either � = 0

16Recall that the torsion subgroup Tor(A) of an abelian group A is the subgroup of A consisting of allelements of finite order.

70

or not. If � = 0, take (1 + pt�)0 = 1. If � 6= 0, then, as Hausdorff, note � 2 pnZp

and distinct from 0 implies � 62 \n�0

pnZp = {0}. Therefore � = ps� for some s � 0

and � 2 Z⇥p . Without loss of generality take U = 1 + pt� + pNZp = 1 + pt+s� + pNZp

for N > t + s. Let d 2 Z, d ⌘ � mod pNZp (so p - d). Then note that this, by a

coset argument, implies U = 1 + pt+sd + pNZp (merely replace coset rep � with d).

Also, let g 2 Z, g ⌘ � mod pNZp, (so p - g). Recall that (Z/pNZ)⇥ = µ(N)p�1 · µ

(N)pN�1

(which = Cp�1 · CpN�1). Then we find < 1 + ptg + pNZ >= µ(N)pN�t as (1 + ptg)k =

1 + pNg + . . . if and only if k � pN�t, or else as |1 + ptg + pNZ| = pN�t because

1 + ptg + pNZ = {a 2 (Z/pNZ)⇥ : apN�t= 1}. Hence we have a generator for µ(N)

pN�t .

Now note that since 1 + pt+sd + pNZ 2 µ(N)pN�t (simply raise to the power N � t),

there exists 2 N such that 1 + pt+sd + pNZ ⌘ (1 + ptg) mod pN . Therefore

(1 + ptg) 2 1 + pt+sd + pNZ ✓ 1 + pt+sd + pNZp = U , a coset equivalent to that

resulting by replacing d with �, as � ⌘ g(pNZp), which yields (1 + pt�) ⌘ (1 + ptg)

mod pNZp. Thus we have 1 + pt� 2 U and so < 1 + pt� > is dense in 1 + ptZp.

Proposition A.24. Given natural numbers M,N with 1 M N . Then (1 + pMZp :

1 + pNZp) = pN�M .

Proof. Since17 the multiplicative factor group Z⇥p /1 + pNZp is algebraically and topo-

logically isomorphic to (Z/pNZ)⇥, which is cyclic, it follows that, as a multiplicative

factor group, (1 + pMZp)/(1 + pNZp) is a finite cyclic subgroup, every element of

which has order pN�M . Let us be cautious here: As a multiplicative factor group,

every element has the form z(1 + pNZp), where z = 1 + pM� with |z| | pN�M . Notice

that, taking � = 1, |(1+ pM)(1+ pNZp)| = pN�M . Hence, the order of the factor group

is precisely pN�M .

Proposition A.25. The sequence 1 ! 1 + psZp ,! Z⇥p

'! (Z/psZ)⇥ ! 1, with

(an + pnZ)n'7! as + psZ, is exact.

17By exact sequence Z⇥p

! (Z/pNZ)⇥ ! 1 with (an

+ pnZ)n

7! aN

+ pNZ.

71

Proof. By compatibility, am ⌘ an(pn), m n, p - a1 (and so p - an for every n 2

N). ' is clearly a well-defined homomorphism. It is surjective, as seen by considering

elements (as + pnZ)n for fixed as. For determining the kernel of ', let ↵ 2 Z⇥p , ↵ =

(an + pnZ)n. Then ↵ 2 ker' if and only if as ⌘ 1 mod ps, if and only if

an ⌘

8

>

>

<

>

>

:

1 mod pn : n s

1 mod ps : n > s

.

Proposition A.26. If H is a closed subgroup of Z⇥p , then H = µf for some f |p � 1 or

H = µf · (1 + psZp) for some f |p� 1 and s 2 N.

Proof. Certainly µf is closed as a finite subset of Hausdorff space Z⇥p , and as clearly

µf · (1 + psZp) = [

!2µf

! · (1 + psZp) is open, hence closed. So we seek to show that

there are no other closed subgroups.

Toward that end, note that either (1 + p)ps�12 H for some s 2 N or not and

consider each case.

Case 1: Suppose (1 + p)ps�12 H for some natural number s, where s is taken

to be minimal. Surely < 1 + p >ps�1✓ H . Taking the closure of each set, note

< 1 + p >ps�1✓ H = H .

Recall that < 1 + p >ps�1= 1 + psZp and consider the factor groups (1) ✓

H/(1+psZp) ✓ Z⇥p /(1+psZp). To show a bijection exists between sets {H : 1+psZp ✓

H ✓ Z⇥p } and {� = H/(1 + psZp) : (1) � Z⇥

p /(1 + psZp)}, note that it would

follow could we show the sequence

1! 1 + psZp�! Z⇥

p

'! (Z/psZ)⇥ ! 1

72

to be exact. But this is the content of Proposition A.25. Hence Z⇥p /(1 + psZp) '

(Z/psZ)⇥, which is cyclic. Further as (Z⇥p : 1+psZp) = �(ps), the factor group H/(1+

psZp) has order (H : 1 + psZp) = ptf where f divides p � 1 and 0 t s � 1. But

now, as (µf · (1+ps�tZp) : 1+psZp) = ptf ,18 the uniqueness of the order of a subgroup

of a cyclic group forces H = µf · (1 + ps�tZp).

Case 2: Suppose (1 + p)pt62 H for every natural number t. I.e., 1 + ptZp 6✓ H

for any t 2 N. Now if H 6✓ µp�1, then, for some h 2 H there would exist ! 2 µp�1

and � 2 Z⇥p such that h = !(1 + ps�). Then hp�1

= !p�1(1 + ps�)p�1

= 1(1 +

ps�)p�1= 1 + (p � 1)ps� + terms with higher powers of p 2 1 + psZp, contradicting

our supposition.

A.8 Dedekind’s Results

We now wish to consider the lattice structure of the closed subgroups and

construct the corresponding subfield lattice as per the generalized Galois correspon-

dence (as per Krull) of extension Q(⇣p1) = [nQ(⇣pn), where19 ⇣pt = e2⇡ipt and p is an

odd prime in N and relate these to the characterizations of elements of these subgroups,

presented as a supplement to the article in the Collected Works and attributed to

Dedekind’s Nachlass.

Toward that end, recall Z⇥p = µp�1 · (1 + pZp), an internal product (p odd),

with µp�1 = {! 2 Z⇥p : !

p�1= 1} ' Cp�1, where ! = !(a) = (ap

n�1+ pnZ)n for

a ⌘ 1, . . . , p � 1 mod p. Also recall, µp�1 ' (Z/pZ)⇥ and ! = !(a) holds precisely

when ! 2 a + pZ. Further, all closed subgroups H have form µf or µf · (1 + psZp),

where p� 1 = ef .18As an internal direct product, we need but verify, for t < s, that ((1 + psZ

p

) : (1 + ptZp

)) = ps�t.But this is immediate by noting (from above) that (1 + p)p

t�1

is a generator for 1 + p 2 Zp

, thus((1 + p)p

t�1

)

p

s�t

2 1 + psZp

, though no lesser power than s� t is contained in 1 + psZp

.19Note, as ↵ 2 Z⇥

p

, ↵ = (an

+ pnZp

), that ⇣↵p

N

= ⇣aN

p

N

.

73

Z⇥p

µp�1 · (1 + p2Zp)

µp�1 · (1 + psZp)

µp�1

µf · (1 + pZp)

µf · (1 + p2Zp)

µf · (1 + psZp)

µf

1 + pZp

1 + psZp

1 + psZp

< 1 >

A1 = Q = B0

A2 = Bp

As = Bps�1

A = Bp1

Qe

QeBp

QeBps�1

QeBp1

Q(⇣p) = P1

Q(⇣p2) = P2

Q(⇣ps) = Ps

Q(⇣p1) = M

p

p

p

e

e

f

f

p

p

p

e

e

e

f

f

f

Figure A.8. Closed subgroup (left) and corresponding field extension (right) sublatticesfor Q(⇣p1) and etc., for fixed e|�(p)

74

Now noting the key structural fact that Z⇥p /(1 + psZp) ' (Z/psZ)⇥, so that,

e.g., Gal(Q(⇣p2/Q) ' (Z/p2Z)⇥ ' Cp�1 · Cp (recall here Cp =< 1 + p + p2Z >),

we thus obtain, for each fixed f |p � 1, the corresponding sublattices shown in Figure

A.8. The nomenclature for subfields in the diagram accords with what appears in the

the supplement to Dedekind’s article (cf. Chapter 3, Translation).

Now for any subfield K ✓M , we observe two cases:

Case 1: K ✓ Q(⇣ps) for some s 2 N, taking s to be minimal such satisfying the

condition. I.e., K 6✓ Q(⇣ps�t) for t < s. Hence, by finite Galois theory, K = AsQe.

Case 2: Suppose K 6✓ Q(⇣ps) for every s 2 N. Then [K : Q] = 1, since

otherwise K = Q(a) for some a 2 M would imply a 2 Q(⇣ps) for some s and hence

also K ✓ Q(⇣ps). We claim that A ✓ K: Recall that for every a 2K, a 2 Q(⇣ps) �

Q(⇣ps�1) for some s 2 N. Hence Q(a) ◆ As. Since [K : Q] =1, there exist an 2 K

such that K = [

1n=1Q(an) = [

1n=1Kn, where Q(an) 6✓ Q(an+1) for each n 2 N+; i.e.,

Kn 6✓ Kn+1. Thus we see As ✓ K for every s, a totally ordered chain of fields bounded

by the factors of p� 1. And, as A = [sAs, A ✓ K.

H MH

µf AQe

µf · (1 + psZp) AsQe

Table A.1. Fixed field-closed subgroup correspondence.

Table A.1 summarizes our findings. Note, e.g., µf is the identitatsgruppen von

M for AQe; that is, the closed subgroup for which AQe is the fixed field.20

Finally, we wish to describe the elements of a given closed subgroup H . To that

end, we have the20That AQ

e

is necessarily the corresponding fixed field, consider the below lattices:

75

Proposition A.27. Take arbitrary ↵ = (an+pnZ)n 2 Z⇥p , where by definition, for every

m n we find an ⌘ am mod pm and (a1, p) = 1. Either (1) ↵ fixes AQe (element

wise) if and only if ↵ 2 µf ; or (2) ↵ fixes AsQe if and only if ↵ 2 µf · (1 + psZp).

Proof. For case (1): ↵ 2 µf if and only if

↵ = !(a1), af1 ⌘ 1 mod p

def.() ↵ = (ap

n�1

1 + pnZ)n, af1 ⌘ 1 mod p

() an ⌘ apn�1

1 mod pn, af1 ⌘ 1 mod p

For case (2): First recall that ↵ 2 1 + psZp if and only if as ⌘ 1 mod ps.21 We

now show the

Proposition A.28. ↵ 2 µf · (1 + psZp) if and only if afs ⌘ 1 mod ps.

Proof. ()) Suppose ↵ 2 µf · (1 + psZp). Then ↵ = !(a1)�, where !(a1) = (apn�1

1 +

pnZ)n, af1 ⌘ 1 mod p and � 2 (1 + psZp), of the form � = (bn + pnZ)n such that

bs ⌘ 1 mod ps. Note as ⌘ aps�1

1 bs mod ps ⌘ aps�1

1 mod ps and a1 = 1 + p for

k

K \ F

K

F

KF

Q = A \Qe

Qe

A

Qe

A

Q(⇣p

)

Q(⇣p

)A

ArbitraryGalois

Galois

µf

e

f

Arbitrary

Note KF/F ' K/(K \ F ) and so Gal(Qe

A/A) ' Gal(Qe

/(A \ Qe

)). Now A \ Qe

= Q as A hasp-power subgroups whereas Q

e

has subgroups of order divisors of p � 1. For A ✓ D ✓ M , all Galois,with [M : A] = p � 1 and e.g. [A : D] = e, immediately the uniqueness of cyclic subgroups forcesD = Q

e

A.21Here is the first of the Dedekind Identity groups, with f = 1. But it is miswritten in the Collected

Works and should instead read (to show compatibility): (n, ") ⌘ (n+ 1, ") mod pn.

76

some 2 Z. Thus

afs ⌘ (aps�1

1 )

fmod ps

⌘ (af1)ps�1

mod ps

⌘ (1 + p)ps�1

mod ps

⌘ 1 mod ps (Binomial theorem.)

(() Suppose22 afs ⌘ 1 mod ps. Consider !(a1) as given above. Since afz ⌘ 1

mod p, !(a1) 2 µf . To show the existence of some satisfactory ↵, consider ↵!(a1)

= � =

(bn + pnZ)n. By consideration of compatibility conditions, it suffices to show bs ⌘ 1

mod ps. Since bs =

as

aps�1

1

, we have (aps�1

1 bs)f⌘ afs mod ps. Now bfs ⌘ 1 mod ps

follows immediately, since af ⌘ 1 mod p implies af1 ⌘ 1 mod ps.23

Observe in particular (for s = 1) aps�1

1 bs ⌘ as mod ps (by compatibility,

again), implying a1b1 ⌘ a1 mod p or b1 ⌘ 1 mod p as (a, p) = 1. Hence bs ⌘ b1 ⌘ 1

mod p, so setting bs = 1+p for some 2 Z we find bps�1

s ⌘ (1+p)ps�1⌘ 1 mod ps

(using the binomial theorem, again). Therefore (f, ps�1) = 1 and bfs ⌘ bp

s�1

s ⌘ 1

mod ps imply bs ⌘ 1 mod ps.

As both cases have been proven, the proposition follows.

22This is the second of the first set of Identity group conditions, namely (s, ")f ⌘ 1 mod ps.23In detail, ap

s�1

bs

⌘ as

mod ps and af1 ⌘ 1(p) imply afps�1

1 ⌘ 1 mod ps. Therefore bfs

⌘

afps�1

1 bfs

⌘ afs

⌘ 1(ps).

77

BIOGRAPHY OF THE AUTHOR

Joseph J.P. Arsenault, Jr. was born in Putnam, Connecticut on April 15, 1964.

He graduated from Wiscasset High School (Wiscasset, Maine) in 1982. He received

a Bachelor of Arts degree in Mathematics from University of Maine in 1995. He is a

candidate for the Master of Arts degree in Mathematics from the University of Maine in

August 2015.

78